Executive Summary

This comprehensive guide examines seven major types of superconducting qubits that have shaped the evolution of quantum computing. Each qubit type represents different optimization priorities in the landscape of superconducting quantum computing. Understanding their designs, advantages, and limitations provides essential insight into the current state and future directions of superconducting quantum computing technology.

Table Of Contents

1. Introduction
2. What Are Charge Qubits?
3. What Are Phase Qubits?
4. What Are Flux Qubits?
5. What Are Transmon Qubits?
6. What Are Fluxonium Qubits?
7. What Are Cat Qubits?
8. What Are 0-π Qubits?
9. Final Thoughts
10. Appendix

1. Introduction

Superconducting qubits have emerged as one of the most promising platforms for building scalable quantum computers, with coherence times improving from nanoseconds to milliseconds over the past two decades [95, 135]. These synthetic atoms, built from macroscopic circuit elements operating at near absolute zero temperatures, offer unprecedented control over quantum states while leveraging decades of semiconductor fabrication expertise [1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 26, 30, 58, 60, 111, 115, 127].

The steady progress in understanding and optimizing superconducting qubits, combined with their fundamental advantages—including fast gate speeds, natural integration with classical electronics, and compatibility with established manufacturing processes—positions superconducting qubits as a cornerstone technology for the quantum computing revolution [139]. Further, the achievement of quantum supremacy, quantum error correction below threshold, and coherence times extending to milliseconds, represent crucial milestones validating superconducting qubits as a viable path toward practical quantum computing [3, 13, 30, 133].

2. What Are Charge Qubits?

Introduction

The charge qubit, also known as a Cooper-pair box (CPB) or C-JJ circuit, was among the first superconducting qubits demonstrated [5, 12, 13, 58, 60, 100, 101]. The Cooper box has been described as being “to quantum circuit physics what the hydrogen atom is to atomic physics” due to its fundamental role and the fact that its eigenenergies and eigenfunctions can be calculated with arbitrary precision using Mathieu functions [102].

Historical Context

Charge qubits originated from Soviet theoretical work on single-electron tunneling, with Kulik and Shekhter’s 1975 paper laying the foundation and Averin and Likharev’s 1986 “Orthodox Theory” providing the complete mathematical framework for Coulomb blockade in small tunnel junctions. Bell Laboratories achieved the first experimental observation of single-electron charging in 1987, validating these theories, while Shnirman, Schön, and Hermon proposed using Cooper pair boxes as quantum bits in 1997. The field transformed in 1999 when Nakamura’s team at NEC demonstrated the first coherent control of a charge qubit with mere nanosecond coherence times, followed by rapid improvements including Vion and Devoret’s 2002 “quantronium” achieving microsecond coherence at operational sweet spots [165]. 

The revolutionary transmon design introduced by Koch and Schoelkopf at Yale in 2007 exponentially suppressed charge noise sensitivity while maintaining qubit control, becoming the dominant architecture for superconducting quantum processors. Modern developments have pushed charge qubits to remarkable performance levels, from Google’s 2019 quantum supremacy demonstration with 53 transmons to MIT’s 2025 achievement of 99.998% single-qubit gate fidelity in fluxonium variants, while innovations in materials like germanium and carbon nanotubes continue expanding the platform’s capabilities toward fault-tolerant quantum computing with processors now containing hundreds to thousands of qubits [165].

Design

At its core, a charge qubit consists of a tiny superconducting island that is coupled to a superconducting reservoir through a Josephson junction [97, 98, 100]. The Josephson junction serves dual purposes, acting simultaneously as both a superconducting tunneling element and a capacitor [60, 101]. The quantum state of this device corresponds to the number of Cooper pairs that tunnel across the junction [13]. This creates a system where quantum information is encoded in charge states [100]. 

Advantages

Unlike the charge states found in atomic or molecular ions, the charge states in these superconducting islands involve a macroscopic number of conduction electrons. This macroscopic nature makes them particularly interesting for quantum computing applications [14, 16, 20, 27, 28, 31, 32, 33, 34, 35, 46, 47, 48, 49, 63, 100].

Disadvantages

However, while charge qubits demonstrated the first temporal coherence in superconducting circuits and featured large anharmonicity (>10 GHz), they suffered from extreme sensitivity to environmental charge noise, limiting their coherence to about 1 microsecond even at optimal operating points –  insufficient for scalable quantum computing applications [3, 5, 13, 58, 119, 127, 128].

3. What Are Phase Qubits?

Introduction

Phase qubits are current-biased circuits where the phase difference across the Josephson junction serves as the primary degree of freedom for encoding quantum information [60, 114, 120, 127].

Historical Context

Phase qubits gained theoretical grounding through Leggett’s 1980 work on macroscopic quantum systems and Caldeira-Leggett’s 1983 framework for understanding decoherence, leading to Martinis, Devoret, and Clarke’s landmark 1985 observation of quantized energy levels in Josephson junctions. After the first superconducting qubit demonstration in 1999, Martinis’s group at UCSB achieved coherent control of phase qubits in 2002 with Rabi oscillations and 1 microsecond coherence times, which they improved 20-fold by 2005 through identifying and mitigating dielectric loss. The technology reached its zenith between 2006-2012 as researchers demonstrated quantum entanglement, Bell inequality violation, three-qubit entangled states, and even implemented Shor’s algorithm to factor 15 with 48% success [165].

However, phase qubits’ fundamental limitations—particularly their susceptibility to current noise and shorter coherence times compared to transmons—led to their abandonment by 2013 when Martinis’s own group introduced the superior Xmon architecture with 44 microsecond coherence times, marking the definitive end of phase qubit development as the entire field transitioned to transmon-based designs [165].

Design

The phase qubit employs a large Josephson junction (EJ/EC ≈ 10⁶) biased by an external current near the critical current [13, 58, 95, 96, 102, 106, 111, 112, 113]. This creates a tilted-washboard potential with weakly anharmonic energy levels [95, 96, 102, 106, 111]. The Josephson junction acts as a non-linear inductor with negligible energy loss, while the capacitance can come from the tunnel junction itself or an external element. This inductor-capacitor resonator exhibits non-linearity even at the single photon level, allowing the two lowest quantum eigenstates to be identified as the qubit states |0⟩ and |1⟩ [15, 29, 36, 37, 39, 59, 113].

Advantages

The large capacitance of phase qubits makes them relatively insensitive to charge noise compared to other qubit designs [113].

Disadvantages

Performance insufficient for scalable quantum computing: phase-slip qubits face several challenges and coherence times remain significantly shorter than relaxation times, indicating strong dephasing mechanisms [113].

4. What Are Flux Qubits?

Introduction

For applications requiring larger anharmonicity than that of charge qubits, the flux qubit was developed [2]. Also known as a “persistent current qubit”, a flux qubit uses persistent currents flowing in opposite directions around a superconducting loop as its two fundamental quantum states [58, 104, 143, 145]: the fundamental quantum mechanical principle governing flux qubits is that only an integer number of flux quanta can penetrate the superconducting ring; this restriction results in the formation of mesoscopic supercurrents that flow either clockwise or counter-clockwise within the loop to compensate for non-integer external flux bias [104].

Historical Context

The late 1990s saw the first detailed theoretical proposals for flux qubits from groups led by Orlando, Mooij, and others, describing three-junction designs with circulating persistent currents as computational states. The year 2000 marked a breakthrough with van der Wal and Friedman’s teams independently demonstrating quantum superposition of macroscopic persistent-current states. Early 2000s developments included the first coherent quantum dynamics and Rabi oscillations (2003), coupling to harmonic oscillators establishing circuit QED principles (2004), and various coupling schemes (2005-2007) [165].

But, the invention of fluxonium in 2009 by Manucharyan introduced a charge-insensitive flux qubit variant using large inductances. Major coherence improvements followed, with capacitively-shunted flux qubits reaching T₁ > 40 microseconds in 2016, fluxonium achieving millisecond coherence in 2019, and gate fidelities steadily improving to 99.8% single-qubit (2021) and 99.92% two-qubit operations (2023). Recent advances include zero-field operation using ferromagnetic π-junctions (2024) and world-record 99.998% single-qubit gate fidelity achieved at MIT in 2025, alongside wafer-scale manufacturing with nearly 100% yield producing qubits with T₁ > 1 millisecond, establishing flux qubits as a mature and highly competitive platform for quantum computing [165].

Design

The flux qubit consists of a micrometer-sized superconducting loop interrupted by three Josephson junctions [2, 4, 6, 7, 8, 13, 58, 60, 104, 105, 107, 108, 143]. The loop encloses a magnetic flux that is carefully controlled and maintained close to half a flux quantum (Φ₀/2), where Φ₀ = h/2e is the fundamental unit of magnetic flux [143]; the junctions are thin insulating barriers that allow quantum tunneling of Cooper pairs (paired electrons in superconductors). Two of the junctions are nearly identical in size and have equal Josephson coupling energy (EJ), while the third junction has a modified coupling energy (αEJ) and a smaller area by a factor α (typically around 0.75-0.5 or less) [2, 107, 108, 143]. This asymmetry is crucial to the qubit’s operation—it controls the qubit’s anharmonicity and creates a double-well potential energy landscape that distinguishes flux qubits from other superconducting qubit designs like phase qubits, which exhibit single-well potentials. Modern flux qubit implementations often employ three-dimensional cavity architectures for improved electromagnetic environment control. These 3D cavities offer well-controlled coupling between the qubit and microwave resonators, while avoiding spurious mode interactions common in two-dimensional designs. The cavity approach also reduces surface dielectric losses [107].

Advantages

The combination of large anharmonicity (enabling fast, high-fidelity gates), high persistent currents (facilitating magnetic coupling to other quantum systems), and improving coherence times, positions flux qubits as a compelling alternative to transmon-based architectures [107, 108, 143].

Controlled Gap Energies & Consistent Relaxation Times

Gap energies controlled to within ±0.6 GHz and consistent relaxation times of 15-20 microseconds [108].

Insensitivity To Flux Noise

The flux qubit becomes immune to flux noise at first order, theoretically enabling longer coherence times, at the optimal point [108].

Insensitivity To Charge Offset Noise

Using magnetic fields makes flux qubits less susceptible to charge noise from impurities, a significant source of decoherence in many solid-state quantum systems [143].

Hybrid Quantum Circuit Compatibility

The large magnetic dipole of flux qubits and their strong magnetic coupling makes them excellent candidates for hybrid quantum circuits, allowing efficient transfer of quantum information between isolated quantum systems such as spins in semiconductors or NV centers in diamond [56, 57, 61, 108].

Reproducibility & Scalability

Recent advances have demonstrated remarkable reproducibility in flux qubit fabrication, with consistent coherence times across large batches of devices [108].  And, with sizes on the order of 1 micrometer, flux qubits can be fabricated using conventional electron beam lithography and integrated into large-scale circuits, potentially enabling quantum computers with thousands of qubits [143].

Magnetic Coupling

A crucial feature for quantum computing is the ability to entangle multiple qubits. In flux qubit systems, this is achieved through magnetic coupling. The persistent current in one qubit generates a small magnetic flux that can influence neighboring qubits. A superconducting flux transporter – essentially a closed superconducting loop with high critical current – can be placed between qubits to mediate controlled coupling, enabling two-qubit gate operations essential for quantum algorithms [11, 50, 71, 94, 143].

Huge Anharmonicity & Limited Frequency Crowding

The most significant advantage of flux qubits is their intrinsically huge anharmonicity—the higher energy levels of the system are very far from the qubit transition (with f₁₂ frequencies around 30 GHz). This property means flux qubits behave as true two-level systems, limiting frequency crowding problems that become increasingly problematic as quantum processors scale up. The large anharmonicity also enables flux qubits to be manipulated on much shorter timescales (less than 10 nanoseconds), potentially achieving better gate fidelities than slower architectures [108]. Modern flux qubit implementations maintain strong anharmonicity of 500-910 MHz, significantly higher than the 200-300 MHz typical of transmon qubits [105].

Disadvantages

The persistent currents that characterize the flux qubit system generate enormous magnetic moments—making the energy of each level extremely sensitive to external magnetic flux [105, 108]. While they can achieve relatively long decoherence times at the degeneracy point (where the two qubit states have equal energy), their performance degrades rapidly when operated away from this point. This sensitivity severely restricts their practical applications in quantum computing and quantum simulation experiments [144].

5. What Are Transmon Qubits?

Introduction

The transmon represents a significant advancement in superconducting qubit design, demonstrating exponential suppression of charge noise sensitivity, without becoming more susceptible to either flux or critical current noise [150]. Transmons were the first qubits to demonstrate quantum supremacy, and have emerged as the dominant superconducting qubit architecture in modern quantum processors [26, 30, 93, 99,115, 116, 118, 129, 131, 132, 133, 134, 136, 137, 138, 140, 141, 142]. Currently, most major quantum computing efforts (Google, IBM, Rigetti) use transmon-based designs due to their scalability potential [106].

Historical Context

The transmon qubit emerged in 2007 when Yale University and Université de Sherbrooke researchers proposed a charge-insensitive design that exponentially suppressed the charge noise plaguing earlier superconducting qubits – by increasing the Josephson to charging energy ratio. Yale experimentally demonstrated the first transmon qubits in 2008, achieving microsecond-range coherence times and proving the theoretical predictions. The architecture quickly gained global adoption, with major advances including Yale’s 3D transmon design in 2011 that improved coherence to 10-20 microseconds, IBM’s introduction of the cross-resonance gate for all-microwave control, and Google’s Xmon architecture in 2013 featuring a cross-shaped grounded design. The field reached critical milestones with IBM launching the first public cloud quantum computer in 2016, Google achieving quantum supremacy with their 53-qubit Sycamore processor in 2019, and Princeton’s breakthrough in 2021 using tantalum to exceed 0.3 millisecond coherence times [165].

The evolution from 2022 to 2025 marked the transition toward practical quantum computing, with IBM demonstrating laser annealing for precise frequency tuning, Chinese researchers achieving 503 microsecond coherence, and Google’s Willow processor in 2024 demonstrating below-threshold quantum error correction with 105 qubits. The field reached new heights in 2025 when Aalto University achieved world record coherence times exceeding one millisecond, representing a 67% improvement over previous records, while systems scaled dramatically with China Telecom’s 504-qubit Tianyan processor and IBM’s Heron R2 capable of 5,000 two-qubit gates within coherence time. From initial microsecond coherence in 2008 to millisecond coherence in 2025—a three orders of magnitude improvement—combined with scaling from single qubits to hundreds of qubits with gate fidelities exceeding 99.9%, the transmon qubit has established itself as the foundation for fault-tolerant quantum computing, powering quantum processors at IBM, Google, and institutions worldwide [165].

Design

A transmon consists of a Josephson junction – a nonlinear superconducting element – connected in parallel with a large capacitor, typically 50-100 times larger than the junction’s intrinsic capacitance [117, 119, 150]. The combination of the junction and shunt capacitor forms a nonlinear resonant circuit with discrete energy levels that serve as the qubit states [117]. The transmon’s configuration creates an anharmonic oscillator that operates at microwave frequencies (typically in the 3-6 GHz range [117, 119, 121]) that makes the qubit charge-insensitive, with low charge dispersion – yielding relatively noise-immune qubits [153, 154].

Advantages

The transmon qubit’s success stems from its optimal balance of multiple critical factors: charge noise insensitivity, high gate fidelities, good connectivity in 2D lattices, material flexibility, structural simplicity, compatibility with standard fabrication techniques, post-fabrication optimization capabilities, reliable reproducibility across many qubits, and sufficient coherence times for deep circuits [22, 153, 154].

Selective Tunability

The ability to selectively tune individual qubit frequencies to desired patterns enables scaling while maintaining high gate fidelities [153].

All-Microwave Control

The transmon architecture supports all-microwave control schemes, eliminating the need for flux control lines in fixed-frequency implementations. This approach reduces the complexity of control electronics, minimizes sources of noise and decoherence, enables faster gate operations, and simplifies cryogenic wiring requirements [153].

Structural Simplicity

The fabrication of transmons employs minimal processes and limited materials complexity – requiring only two lithography layers on a single-crystal sapphire or silicon substrate, and a single cavity – for reliable control and readout, as well as reproducible qubit coherence and relaxation times [17, 36, 51, 52, 53, 54, 55, 149, 150].

Dispersive Readout

Transmons naturally support dispersive readout through coupling to microwave resonators. This readout mechanism provides high-fidelity qubit state discrimination, allows for quantum non-demolition measurements, enables multiplexed readout of multiple qubits, and maintains good isolation between qubits during idle periods [154].

Scalable Fabrication

The transmon design is particularly well-suited for scalable fabrication using dry etching processes, which offer advantages over wet etching such as high anisotropy for precise pattern definition, automation in large-scale production, reduced material consumption, better industrial hygiene and process control [154], and compatibility with standard semiconductor manufacturing infrastructure – but without deposited dielectric layers [127, 130, 149, 150, 154].

Diverse Material Options

Flexibility in material choice, from traditional aluminum and niobium to emerging platforms like tantalum, ensures continued transmon performance improvements [154]. Combined with advanced techniques like laser annealing for frequency allocation and dry etching processes suitable for large-scale manufacturing, transmon qubits represent a mature, yet still rapidly advancing, technology platform for quantum computing [153, 154].

Suppression Of Charge Dispersion 

As the E_J/E_C ratio increases, the energy levels of the transmon qubit become increasingly flat as a function of gate charge, effectively creating what researchers describe as a “sweet spot everywhere” [128]. This flattening indicates strong suppression of charge dispersion—the variation of energy levels with offset charge—which is the transmon’s key advantage over earlier designs like the Cooper Pair Box [139]. With suppressed charge dispersion, the dephasing time for the transmon is substantially improved [149, 150]. 

Insensitivity To Charge Noise

The key innovation of the transmon design is the exponential suppression of sensitivity to charge noise – random fluctuations in the local electrostatic environment that plagued conventional charge qubits – with sensitivity to charge noise reduced by approximately eight orders of magnitude [103, 117, 127, 128, 150, 154]. This is achieved by dramatically increasing the E_J/E_C ratio from ~0.1 to ~100, primarily by adding to the circuit a large shunt capacitor in parallel with the Josephson junction [1, 2, 3, 13, 95, 103, 127, 128, 150, 153]. This takes the majority of the charge noise effect, minimizing its impact on the junction (“shunting the junction”) [1, 2, 3, 5, 13]. Advantageously, while transmon’s sensitivity to charge noise decreases exponentially with E_J/E_C, anharmonicity decreases only algebraically, following a weak power law [95, 119, 127, 128, 150]. 

Disadvantages

Despite remarkable progress, transmon qubits face ongoing challenges, such as crosstalk, frequency crowding, and the need for extensive calibration in multi-qubit systems. Further, each transmon typically couples directly to only a few neighboring qubits, requiring careful design of coupling networks. And, microscopic two-level defects, electromagnetic interference, and residual thermal photons continue to cause energy loss or phase errors [19, 117, 119]. Finally, transmon qubits are challenged by leakage to higher excited states beyond the computational basis. The importance of leakage mitigation is demonstrated by a 35% improvement in the error suppression factor (Λ) when DQLR is activated for distance-5 codes [133].

6. What Are Fluxonium Qubits?

Introduction

The fluxonium qubit offers a unique solution to the challenge of maintaining quantum coherence while preserving strong anharmonicity, with exceptional coherence properties, tunable energy spectrum, and robust protection against environmental noise [109, 110, 122, 123, 124, 125, 126, 135, 151, 152]. Recent advances have pushed fluxonium performance to remarkable levels, enabling coherence times exceeding 1 millisecond [95, 115, 135].

Historical Context

The experimental history of fluxonium qubits began in 2009 when Vladimir E. Manucharyan and colleagues at Yale University first demonstrated these novel superconducting circuits, achieving complete insensitivity to charge noise while maintaining single Cooper pair effects. Early milestones included observing coherent quantum phase slips in 2012 and discovering π-phase suppression of quasiparticle dissipation in 2014, which enhanced coherence times from microseconds to over 40 microseconds. The field progressed through demonstrations of artificial molecules and protection mechanisms before achieving a major breakthrough in 2019 – when the University of Maryland team reproducibly exceeded 100 microseconds of coherence, with some devices reaching over 400 microseconds [165].

The early 2020s saw rapid global advancement, with Alibaba’s quantum laboratory achieving 99.97% single-qubit and 99.72% two-qubit gate fidelities, Russian researchers demonstrating 99.55% fSim gates, and MIT developing hybrid fluxonium-transmon architectures with 99.9% CZ gate fidelities. The field reached transformative milestones in 2023 when University of Maryland researchers broke the millisecond coherence barrier with 1.48 millisecond Ramsey times, followed by demonstrations of 24-day stable CNOT gates above 99.9% fidelity and wafer-scale manufacturing with nearly 100% yield in 2024. The culmination came in January 2025 when MIT achieved a world record 99.998% single-qubit gate fidelity using novel control techniques, establishing fluxonium as the leading superconducting qubit platform for fault-tolerant quantum computing [165].

Design

A fluxonium qubit consists of three fundamental components working in harmony: a Josephson junction, a large shunting inductor, and a shunting capacitor [109, 110, 122, 123, 124, 125, 126, 151, 152]. The defining parameters of fluxonium satisfy EL ≪ EJ and 1 ≲ EJ/EC ≲ 10, which distinguish it from other inductively-shunted junction devices [135, 152]. The fluxonium qubit spectacularly breaks the conventional wisdom that junction count must be minimized for optimal coherence [152]: by employing over 100 junctions, it achieves simultaneous protection against both dielectric loss and flux noise without sacrificing anharmonicity or controllability [151, 152]. For successful implementation of a charge offset-free inductively shunted junction, four critical conditions must be met: the total array inductance must exceed the small junction inductance to support large flux fluctuations, array junctions must be large enough to reduce uncontrolled offset charges, phase slip probability in the array must be negligible compared to the small junction, and array inductance must not be shunted by parasitic capacitances to ground [151]. Fluxonium qubits are coupled to readout resonators through capacitive coupling [152].

Advantages

The unique combination of long coherence time, large anharmonicity, exceptional noise protection, and compatibility with various coupling schemes positions fluxonium as a versatile platform for both gate-based and adiabatic quantum computing [109, 110, 122, 123, 124, 125, 126, 152].

All-Microwave Control

Fluxonium’s large anharmonicity at the sweet spot allows for fast local microwave control and better scalability [115, 152].

Compatibility

Fluxonium qubits have been implemented in devices simultaneously utilizing capacitively coupled transmons, thus demonstrating scalability and compatibility with hybrid quantum systems [125, 126]. Further, fluxonium qubits demonstrate strong dispersive coupling (690 MHz) with mechanical resonators, enabling integration with optomechanical systems [125].

Insensitivity To Flux Noise

The fluxonium qubit’s large shunting inductance provides extraordinary protection against flux noise and, unlike flux qubits, fluxonium shows no signatures of flux-noise-induced energy relaxation [152]. At the commonly used half-integer flux quantum bias (the “sweet spot”), the fluxon states degenerate, creating a first-order flux-noise-insensitive point [95, 115, 135, 152].

Insensitivity To Charge Offset Noise

The single-island design of fluxonium qubits makes them immune to noise from offset charges – the random, slowly drifting, microscopic charges that severely limited the coherence of earlier superconducting qubits like the Cooper pair box [95, 115, 135, 151, 152]. This charge-free character distinguishes fluxonium from the transmon, where offset charge influence is only screened for low-lying states [151].

Quantum Adiabatic Optimization

Networks of fluxoniums can implement generic quantum spin-1/2 Hamiltonians with unprecedented coherence for quantum annealing – a quantum computing approach that uses gradual quantum evolution to solve optimization problems by encoding them into a landscape of energy states (a quantum Hamiltonian) and guiding the system to its lowest-energy state (the ground state), which corresponds to the optimal solution [24, 25, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 152].

Disadvantages

One primary challenge in operating fluxonium qubits arises from their low transition frequency. When implementing fast gates with durations approaching the qubit Larmor period, counter-rotating effects become significant and can severely degrade gate fidelity [126].

7. What Are Cat Qubits?

Introduction

Named after Schrödinger’s famous thought experiment, cat qubits leverage the infinite-dimensional Hilbert space of a quantum harmonic oscillator (a bosonic mode) to achieve hardware-efficient, fault-tolerant quantum computation [155, 156, 157, 158, 159, 160]. By encoding information in superpositions of coherent states, and stabilizing them through engineered dissipation, cat qubits achieve exponential suppression of bit-flip errors – while maintaining manageable phase-flip rates [18, 156, 157]. The ability to perform universal quantum computation with a simple repetition code, but without magic state preparation, positions cat qubits as a promising candidate for near-term demonstrations of fault-tolerant quantum computing. Recent experimental progress in achieving high-fidelity operations and observing predicted error suppression validates the theoretical framework, and built-in noise resilience through bias-aware error correction, combined with asymmetric error channels, make cat qubits particularly attractive for practical quantum computing applications [18, 157].

Historical Context

Cat qubits evolved from theoretical curiosity to practical quantum computing technology over nearly three decades. The journey began in 1996 when Haroche’s team at École Normale Supérieure created the first Schrödinger cat states in laboratory cavities, proving macroscopic quantum superpositions could exist. In 1999, Australian physicists Cochrane, Milburn, and Munro proposed using these cat states for quantum error correction, though this idea remained dormant for years. While the early 2000s saw researchers like Gottesman, Kitaev, and Preskill develop the theoretical framework for bosonic quantum computing, the modern cat qubit era truly began in 2013 when Yale and INRIA collaborators introduced cat qubits with autonomous error protection, exploiting their unique biased noise property where bit-flip errors are exponentially suppressed while phase-flips remain manageable [165].

Following theoretical breakthroughs in 2014 showing universal quantum computation was possible, experimental demonstrations accelerated: Yale achieved the first error correction extending qubit lifetime beyond natural limits in 2016, and 2020 brought multiple milestones including the founding of Alice & Bob and experimental proof of exponential bit-flip suppression. The field reached maturity in 2024 when Alice & Bob achieved ten-second bit-flip times—a four-orders-of-magnitude improvement—and AWS launched the first industrial implementation in 2025, marking cat qubits’ transformation from abstract quantum mechanics to commercial reality [165].

Design

Cat qubits encode information in coherent states of light in a harmonic oscillator [18, 155, 156, 157, 158, 159, 160]. Cat qubits are stabilized using engineered two-photon dissipation, which maintains the cat-qubit amplitude α and the noise bias over time [18, 135, 156, 157, 158, 159, 160]. The two-photon dissipation is implemented by coupling each storage mode (as λ/2 coplanar waveguide resonators) to a lossy buffer mode through a three-wave mixing (3WM) Hamiltonian, combined with a linear drive and strong loss on the buffer mode [158, 159, 160]. A critical innovation is the use of 4-pole metamaterial bandpass filters to protect storage modes from buffer loss channels [158, 159]. For control, cat qubits require a high-Q resonator in addition to a nonlinear element or an ancilla qubit [18].

Advantages

Advantages of the cat qubit regime over traditional approaches include drastically reduced overhead requirements, simpler gate implementations, direct non-Clifford gates without distillation, extended gate set including CNOT and Toffoli, no required concatenation for universality, and topological protection without actual topology [40, 41, 42, 43, 44, 45, 62, 157].

Hardware Efficiency

Cat qubit hardware efficiency provides the following benefits: a single oscillator replaces multiple physical qubits, no increase in decay channels with encoding, and tunable protection level via cat size [155, 156, 157].

Simplified Fault Tolerance

Simplified fault tolerance provides the following benefits: universal gates at repetition code level, universality without magic state preparation, and simplified Clifford gate circuits [157].

Extreme Noise Bias

The noise bias of cat qubits is tunable, and can reach values exceeding 10⁸ even for modest photon numbers. This extreme bias arises from the exponential suppression of bit-flip errors combined with a linear increase in phase-flip errors (which are caused by single-photon loss and heating) [18, 156, 157, 158, 159, 160].

Exponential Bit-Flip Suppression

The construction of cat qubits circumvents the no-go theorem for bias-preserving gates, enabling a universal set of bias-preserving gates that maintain exponential bit-flip suppression – where measured values approach the theoretical limit expected under white-noise dephasing – by exploiting the infinite-dimensional Hilbert space [9, 10, 156, 157, 158, 160]. Transitioning between distinct states in phase space is extremely unlikely – the states are separated enough that they rarely overlap, making accidental transitions exponentially suppressed as the cat states become more macroscopic (larger α values) [18, 157]. Bit-flip errors are exponentially suppressed for any noise process with local effect in phase space – including photon loss, thermal excitations, photon dephasing, and Josephson nonlinearities [156, 157].

Disadvantages

There are several technical challenges to the cat qubit approach, including maintaining high κ₂ₚₕ/κ₁ₚₕ ratios, scaling to many cat qubits, cross-talk management in multi-qubit systems, optimal decoder implementation, and parameter drift management [18, 157]. Precise calibration requirements for stabilizing cat states introduce challenges with parameter drift that must be carefully managed in order to avoid additional noise or failure modes [18].

8. What Are 0-π Qubits?

Introduction

The 0-π qubit is a superconducting quantum circuit designed to achieve hardware-level error suppression in superconducting quantum computers [146, 147, 148, 161, 162, 163, 164]. A rudimentary form of topological protection that combines exponential suppression of noise-induced transitions (dissipation) with exponential suppression of dephasing [162, 163], it is through a novel combination of disjoint support wave functions and near-degenerate energy levels that 0-π qubits are distinguished from conventional superconducting qubits [146, 147, 148, 161, 162, 163, 164]. The concatenation of transmon-like (θ) and fluxonium-like (φ) modes provides more protection than possible with a single mode, combining the best features of both qubit types [164].

Historical Context

The history of 0-π qubits traces back to foundational work in 2001 when Gottesman, Kitaev, and Preskill established the GKP code framework for continuous-variable quantum error correction, followed by Ioffe and colleagues’ 2002 introduction of topological protection concepts using Josephson junction arrays. The conceptual blueprint emerged in 2006 when Kitaev proposed the protected qubit based on a superconducting current mirror with exponentially protected qubits through exciton condensation. The formal introduction of 0-π qubits came in 2013 when Brooks, Kitaev, and Preskill demonstrated quantum phase gates protected by continuous-variable quantum error-correcting codes, showing that a cos(2φ) Josephson potential creates a double-well structure enabling protected logical states with exponentially small gate errors. Theoretical analysis in 2018 by Groszkowski and collaborators predicted that 0-π qubits could surpass the best superconducting qubits with proper optimization, leading to the first experimental demonstration in 2019 by the Princeton team who created a “soft 0-π qubit” that maintained noise protection in an experimentally accessible regime [165].

Related work in 2020 on the bifluxon demonstrated a 10-fold increase in relaxation time with activated protection, and the Princeton group published further results in 2021 achieving relaxation times of 1.6 ms and echo dephasing times of 25 μs. Recent advances include asymmetric control methods in 2023, the 2024 achievement of the world’s first superconducting flux qubit operating without magnetic field using ferromagnetic π-junctions with a 360-fold improvement over previous implementations, and a 2025 theoretical solution to the controllability problem through a novel protected phase gate design, addressing a major practical hurdle for implementing 0-π qubits in quantum computing applications [165].

Design

The 0-π qubit consists of four nodes connected in a closed-loop geometry with carefully chosen circuit elements: a pair of linear inductors, a pair of capacitors, and a pair of Josephson junctions [146, 147, 148, 161, 162, 163, 164]. The 0-π circuit has four degrees of freedom, four quadrupole circuit modes – referred to as the φ, θ, Σ, and ζ, modes – which correspond to linear combinations of the phase differences across the various circuit elements [162, 163, 164]. The θ and φ modes encode the qubit [164]; the Σ mode is cyclic, non-dynamical, remains decoupled, and is discarded [162, 163, 164]; the ζ mode is harmonic and ideally remains decoupled from the qubit modes in the absence of circuit-element disorder [147, 148, 162, 163, 164].

Advantages

The 0-π qubit achieves simultaneous protection against both energy relaxation and pure dephasing – but at the cost of many engineering and fabrication challenges [146, 147, 148, 161, 162, 163, 164].

Temperature Robustness

0-π qubit gate performance remains robust at finite temperatures, with protection maintained when temperature is below the oscillator frequency [161].

Topological Protection

The 0-π qubit provides a rudimentary form of topological protection at the hardware level, potentially reducing the overhead required for quantum error correction compared to unprotected qubits [21, 23, 162, 163].

Simultaneous Protection Against Energy Relaxation & Pure Dephasing

The most significant advantage of the  0-π qubit is simultaneous protection against both energy relaxation and pure dephasing – with error rates that decrease exponentially as circuit parameters are improved. This dual protection is unique compared to conventional superconducting qubits, which typically can only optimize for one type of error at a time [146, 147, 148, 161, 162, 163, 164].

Energy Relaxation Protection

The 0-π qubit achieves protection against energy relaxation through wave function localization. Wave function localization in the 0-π qubit means the two computational states (|0⟩ and |π⟩) are spatially separated in phase space, residing in different wells of the double-well potential. Spatial separation exponentially suppresses transition matrix elements between the logical states induced by local noise operators (like charge noise or flux noise), leading to remarkably long T₁ times [146, 147, 148, 164].

Pure Dephasing Protection

Charge noise protection comes from the transmon regime operation. Operating the θ mode in the transmon regime with E^θ_C ≪ E_J exponentially suppresses charge sensitivity [147, 148, 163, 164]. Exploiting the avoided crossing of the two lowest-lying levels (the qubit’s two minima) in the θ = π valley creates a first-order flux sweet spot at zero external flux – providing natural insensitivity to flux fluctuations, but without requiring fine-tuning [147, 163, 164]. The hybridization gap at this sweet spot, proportional to the tunneling rate between local minima in the π valley, requires sufficiently large E^φ_C [163]. 

Disadvantages

0-π qubits face many engineering and fabrication challenges [147, 148, 163, 164].

Low Electrical Resistance

Achieving superinductance L/C ≫ 1 kΩ remains a significant engineering challenge [161, 162]. 

Low Resistance Quantum

Current superinductor performance is limited to ~5 R_Q, with ~50 R_Q needed for full protection [164].

Cooling Requirements

The ancillary mode (whether external oscillator or internal ζ mode) must be cooled between gate operations in order to extract entropy [164].

Quasiparticle Poisoning & Dielectric Losses

Minimizing quasiparticle poisoning and dielectric losses, two major sources of decoherence in superconducting qubits, 0-π qubit continue as fabrication challenges for [163].

Junction Uniformity

Junction uniformity is a practical limitation in 0-π qubits, which require chains of 400+ Josephson junctions or geometric coils to achieve necessary inductance values [164]. 

Parasitic Capacitance

While silicon-on-insulator substrates help reduce stray capacitances that decrease E^φ_C, parasitic capacitance currently prevents achieving ultralight flux mode required for full protection [164].

Consequences Of Increased Qubit Protection

A consequence of increased qubit protection is an increased difficulty to perform quantum gates by conventional means. Rabi-style gates are infeasible for protected qubits since the matrix elements between computational states are small by design. This can be overcome in the near-term by utilizing non-computational states for Raman-style gates, but this strategy is only feasible for 0-π qubits in the partially protected parameter regime, not in the fully protected regime [164].

9. Final Thoughts

The evolution of superconducting qubits from theoretical curiosity to the foundation of commercial quantum processors represents one of the most remarkable achievements in modern physics and materials engineering. Each qubit architecture examined in this guide—from the pioneering charge qubits to the sophisticated cat and 0-π designs—embodies unique solutions to the fundamental challenge of maintaining quantum coherence while enabling precise control. Looking forward, the ultimate winner in this technological race may not be a single architecture, but rather hybrid systems that optimize for the roles and strengths of multiple qubit types.

10. Appendix

Superconducting Qubit Type Advantages & Disadvantages

Glossary Of Key Terms

Anharmonicity – The property of an energy spectrum where energy level spacings are unequal. In qubits, this allows selective addressing of the two lowest energy states without exciting higher levels. Measured in frequency units (MHz or GHz).

Bosonic mode – A quantum harmonic oscillator mode that follows Bose-Einstein statistics, such as photons in a cavity. Can support multiple excitations simultaneously, unlike fermionic modes.

Charge dispersion – The variation of qubit energy levels with offset charge. A key source of decoherence in early superconducting qubits, exponentially suppressed in transmon designs.

Charge noise – Random fluctuations in the local electrostatic environment that cause dephasing in charge-sensitive qubits. Originates from trapped charges in substrate materials and interfaces.

Circuit QED (Quantum Electrodynamics) – The study of quantum phenomena in superconducting circuits coupled to microwave resonators, analogous to cavity QED with atoms.

Coherence time – The duration a qubit maintains quantum superposition before environmental decoherence destroys it. Includes both T₁ (relaxation) and T₂ (dephasing) times.

Cooper pair – A bound state of two electrons with opposite spins and momenta that forms the charge carrier in superconductors. Has charge 2e and zero resistance.

Coulomb blockade – The suppression of electron tunneling at low bias voltages due to electrostatic charging energy in small tunnel junctions.

Dephasing (T₂) – Loss of phase coherence between quantum states without energy exchange. Pure dephasing (T₂*) occurs without relaxation.

Dispersive readout – A quantum non-demolition measurement technique where the qubit state shifts the frequency of a coupled resonator, enabling state discrimination without direct measurement.

EC (Charging energy) – The electrostatic energy required to add one Cooper pair to a superconducting island. Inversely proportional to capacitance.

EJ (Josephson energy) – The characteristic energy scale of a Josephson junction, proportional to the critical current. Determines the coupling strength for Cooper pair tunneling.

EL (Inductive energy) – The characteristic energy scale associated with inductance in a circuit, particularly relevant for flux and fluxonium qubits.

Flux noise – Low-frequency magnetic field fluctuations that cause dephasing in flux-sensitive qubits. Often exhibits 1/f spectrum.

Flux quantum (Φ₀) – The fundamental unit of magnetic flux, equal to h/2e ≈ 2.07×10⁻¹⁵ Wb. Superconducting loops can only enclose integer multiples of flux quanta.

Gate fidelity – The accuracy of a quantum gate operation, typically expressed as a percentage. Measures how closely the implemented operation matches the ideal transformation.

Hamiltonian – The mathematical operator describing the total energy of a quantum system, determining its time evolution and energy eigenstates according to the Schrödinger equation.

Impedance (Z) – The effective resistance of a circuit element to alternating current, combining both resistive and reactive components. In quantum circuits, characteristic impedance Z₀ = √(L/C) determines coupling strengths.

Inductance (Kinetic) – In superconducting circuits, the property arising from the kinetic energy of Cooper pairs that creates an effective inductance without physical coils, crucial for fluxonium and 0-π qubit designs.

Josephson junction – A thin insulating barrier between two superconductors that allows quantum tunneling of Cooper pairs. Acts as both a nonlinear inductor and capacitor.

Kerr nonlinearity – A third-order nonlinear optical effect in quantum systems where the refractive index depends on light intensity, enabling self-phase modulation and creating anharmonicity in superconducting circuits.

Kinetic inductance – Inductance arising from the kinetic energy of charge carriers rather than magnetic field storage. Dominant in thin superconducting films and nanowires, used to create high-impedance elements.

Larmor frequency – The precession frequency of a quantum state around the Bloch sphere, determined by the energy splitting between qubit states.

Macroscopic quantum state – A quantum superposition involving a large number of particles behaving collectively, such as persistent currents in superconducting loops.

Mathieu functions – Special functions that describe the eigenstates of periodic quantum systems, used to calculate exact solutions for charge qubits.

Noise Bias – The ratio between phase-flip and bit-flip error rates in a qubit system; cat qubits achieve extreme bias (>10⁸) by exponentially suppressing bit-flips while maintaining linear phase-flip rates.

Noise spectral density – The distribution of noise power across different frequencies, typically following 1/f (flicker) or white noise patterns. Determines decoherence rates at different operating frequencies.

Offset charge – Random, slowly-drifting background charges trapped in dielectric materials near qubits. Major decoherence source in early charge qubits, suppressed in modern designs.

Persistent current – Macroscopic supercurrent flowing indefinitely in a superconducting loop without applied voltage, used as computational states in flux qubits.

Phase slip – A quantum tunneling event where the superconducting phase changes by 2π, allowing flux quanta to enter or leave a superconducting loop.

Purcell effect – Modification of spontaneous emission rates due to electromagnetic environment, can cause unwanted qubit relaxation through coupled resonators.

Quantum annealing – An optimization algorithm that uses quantum tunneling to find global minima in energy landscapes by slowly evolving from an initial superposition to the ground state of a problem Hamiltonian.

Quantum anharmonic oscillator – A quantum system with unequally spaced energy levels, essential for isolating two-level qubit subspaces from higher excitations.

Quantum non-demolition (QND) – Measurement that determines a quantum property without disturbing it, allowing repeated measurements of the same observable.

Quasiparticle – Unpaired electrons that break Cooper pairs, causing dissipation and decoherence in superconducting circuits. Minimized by operating at millikelvin temperatures.

Rabi oscillations – Coherent oscillations between qubit states under resonant driving, characterized by the Rabi frequency proportional to drive amplitude.

Relaxation time (T₁) – The characteristic time for a qubit to decay from excited to ground state through energy dissipation to the environment.

Superinductance – Extremely large inductance (>100 nH) achieved using arrays of Josephson junctions or kinetic inductance materials, crucial for fluxonium qubits.

Sweet spot – Operating point where qubit frequency is insensitive to first-order noise fluctuations, maximizing coherence times.

Three-wave mixing (3WM) – Nonlinear process where three electromagnetic modes interact, used for parametric amplification and two-photon dissipation in cat qubits.

Transmon – A charge-insensitive superconducting qubit design featuring a large shunt capacitor parallel to a Josephson junction, achieving exponential suppression of charge noise while maintaining sufficient anharmonicity.

Transmon regime – Operating condition where EJ/EC ≫ 1, providing exponential suppression of charge noise sensitivity at the cost of reduced anharmonicity.

Two-level system (TLS) defects – Microscopic defects in materials that act as parasitic qubits, causing decoherence through resonant energy exchange or dephasing.

Ultrastrong coupling – Regime where qubit-resonator coupling strength approaches or exceeds the resonator frequency. Enables novel quantum phenomena but complicates standard approximations.

Vacuum Rabi splitting – The energy splitting observed when a qubit strongly couples to a resonator mode, creating dressed states that are superpositions of qubit and photon states.

Wave function localization – Spatial separation of quantum states in phase space, providing protection against transitions induced by local noise operators.

Xmon – A transmon variant developed by Google with a cross-shaped capacitor design, providing improved connectivity and reduced crosstalk in 2D arrays.

Y-factor – The ratio of noise powers measured at two different temperatures, used to characterize amplifier noise performance in quantum measurement chains.

Zero-point fluctuations – Quantum fluctuations present even at absolute zero temperature due to Heisenberg uncertainty. Sets fundamental noise floor in quantum circuits and determines vacuum impedance.

References

[1] Kwon, et al. – Gate-based superconducting quantum computing – https://pubs.aip.org/aip/jap/article/129/4/041102/957183/Gate-based-superconducting-quantum-computing

[2] Krantz, et al. – A quantum engineer’s guide to superconducting qubits – https://pubs.aip.org/aip/apr/article/6/2/021318/570326/A-quantum-engineer-s-guide-to-superconducting

[3] Kjaergaard, et al. – Superconducting Qubits: Current State of Play – https://www.annualreviews.org/content/journals/10.1146/annurev-conmatphys-031119-050605

[4] Gambetta, et al. – Building logical qubits in a superconducting quantum computing system – https://www.nature.com/articles/s41534-016-0004-0

[5] Murray, C. – Material matters in superconducting qubits – https://arxiv.org/pdf/2106.05919

[6] Oliver & Welander – Materials in superconducting quantum bits – https://www.researchgate.net/publication/259437347_Materials_in_superconducting_quantum_bits

[7] Lu, et al. – Two-qubit gate operations in superconducting circuits with strong coupling and weak anharmonicity – https://iopscience.iop.org/article/10.1088/1367-2630/14/7/073041

[8] Yost, et al. – Solid-state qubits integrated with superconducting through-silicon vias – https://www.nature.com/articles/s41534-020-00289-8

[9] DiVincenzo, D. – The Physical Implementation of Quantum Computation – https://arxiv.org/pdf/quant-ph/0002077

[10] Loss & DiVincenzo – Quantum Computation with Quantum Dots – https://arxiv.org/pdf/cond-mat/9701055

[11] Chae, et al. – An elementary review on basic principles and developments of qubits for quantum computing – https://nanoconvergencejournal.springeropen.com/articles/10.1186/s40580-024-00418-5

[12]  Ivezic, M. – Quantum Computing Modalities: Gate-Based / Universal QC – https://postquantum.com/quantum-modalities/gate-based-universal-quantum/

[13] Ivezic, M. – Quantum Computing Modalities: Superconducting Qubits – https://postquantum.com/quantum-modalities/superconducting-qubits/

[14]  Ivezic, M. – Quantum Computing Modalities: Trapped-Ion QC – https://postquantum.com/quantum-modalities/trapped-ion-qubits/

[15]  Ivezic, M. – Quantum Computing Modalities: Photonic QC – https://postquantum.com/quantum-modalities/photonic-quantum-computing/

[16]  Ivezic, M. – Quantum Computing Modalities: Neutral Atom (Rydberg) – https://postquantum.com/quantum-modalities/neutral-atom-quantum/

[17]  Ivezic, M. – Quantum Computing Modalities: Silicon-Based Qubits – https://postquantum.com/quantum-modalities/silicon-based-qubits/

[18] Ivezic, M. – Quantum Computing Modalities: Superconducting Cat Qubits – https://postquantum.com/quantum-modalities/superconducting-cat-qubits/

[19] Ivezic, M. – Quantum Computing Modalities: Photonic Cluster-State – https://postquantum.com/quantum-modalities/photonic-cluster-state/

[20] Ivezic, M. – Quantum Computing Modalities: Ion Trap and Neutral Atom MBQC – https://postquantum.com/quantum-modalities/ion-trap-neutral-atom-mbqc/

[21] Ivezic, M. – Quantum Computing Modalities: Topological Quantum Computing – https://postquantum.com/quantum-modalities/topological-quantum-computing/

[22] Ivezic, M. – Quantum Computing Modalities: Majorana Qubits – https://postquantum.com/quantum-modalities/majorana-qubits/

[23] Ivezic, M. – Quantum Computing Modalities: Fibonacci Anyons – https://postquantum.com/quantum-modalities/fibonacci-anyons/

[24] Ivezic, M. – Quantum Computing Modalities: Quantum Annealing (QA) & Adiabatic QC (AQC) – https://postquantum.com/quantum-modalities/annealing-adiabatic/

[25] Ivezic, M. – Quantum Computing Modalities: Quantum Annealing (QA) – https://postquantum.com/quantum-modalities/quantum-annealing/

[26] Grumbling & Horowitz – Quantum Computing: Progress and Prospects – https://nap.nationalacademies.org/read/25196/chapter/1

[27] Liu, et al. – Certified randomness using a trapped-ion quantum processor – https://www.nature.com/articles/s41586-025-08737-1

[28] Ballon, A. – Trapped ion quantum computers – https://pennylane.ai/qml/demos/tutorial_trapped_ions

[29] Ballon, A. – Photonic quantum computers – https://pennylane.ai/qml/demos/tutorial_photonics

[30] Ballon, A. – Quantum computing with superconducting qubits – https://pennylane.ai/qml/demos/tutorial_sc_qubits

[31] Killoran, N. – Quantum computation with neutral atoms – https://pennylane.ai/qml/demos/tutorial_pasqal

[32] Ballon, A. – Neutral-atom quantum computers – https://pennylane.ai/qml/demos/tutorial_neutral_atoms

[33] Wintersperger, et al. – Neutral Atom Quantum Computing Hardware: Performance and End-User Perspective – https://arxiv.org/pdf/2304.14360

[34] Swayne, M. – Harnessing The Power Of Neutrality: Comparing Neutral-Atom Quantum Computing With Other Modalities – https://thequantuminsider.com/2024/02/22/harnessing-the-power-of-neutrality-comparing-neutral-atom-quantum-computing-with-other-modalities/

[35] Wikipedia – Neutral atom quantum computer – https://en.wikipedia.org/wiki/Neutral_atom_quantum_computer

[36] Romero & Milburn – Photonic Quantum Computing – https://docs.google.com/document/d/1bZbxz1ZSZDN2rW71GzDQbK4NHeWVhxcQ7wU9COo91_U/edit?tab=t.0

[37] Wikipedia – Linear optical quantum computing – https://en.wikipedia.org/wiki/Linear_optical_quantum_computing

[38] Kramnik, et. al – Scalable Feedback Stabilization of Quantum Light Sources on a CMOS Chip – https://arxiv.org/pdf/2411.05921

[39] Rad, et. al – Scaling and networking a modular photonic quantum computer – https://www.nature.com/articles/s41586-024-08406-9

[40] Nayak, C. – Microsoft unveils Majorana 1, the world’s first quantum processor powered by topological qubits – https://azure.microsoft.com/en-us/blog/quantum/2025/02/19/microsoft-unveils-majorana-1-the-worlds-first-quantum-processor-powered-by-topological-qubits/

[41] Microsoft Azure Quantum, et. al – Interferometric single-shot parity measurement in InAs–Al hybrid devices – https://www.nature.com/articles/s41586-024-08445-2

[42] Microsoft Quantum – InAs-Al hybrid devices passing the topological gap protocol – https://journals.aps.org/prb/pdf/10.1103/PhysRevB.107.245423

[43] Microsoft Quantum – Roadmap to fault tolerant quantum computation using topological qubit arrays – https://arxiv.org/pdf/2502.12252

[44] Chao, et. al – Optimization of the surface code design for Majorana-based qubits – https://quantum-journal.org/papers/q-2020-10-28-352/pdf/

[45] Wikipedia – Topological quantum computer – https://en.wikipedia.org/wiki/Topological_quantum_computer

[46] Iqbal, et. al. – Qutrit Toric Code and Parafermions in Trapped Ions – https://arxiv.org/html/2411.04185v1#S6

[47] Iqbal, et. al. – Non-Abelian topological order and anyons on a trapped-ion processor – https://arxiv.org/pdf/2305.03766

[48] Mayer, et. al – Benchmarking logical three-qubit quantum Fourier transform encoded in the Steane code on a trapped-ion quantum computer – https://arxiv.org/pdf/2404.08616

[49] Ryan-Anderson, et. al – High-fidelity and Fault-tolerant Teleportation of a Logical Qubit using Transversal Gates and Lattice Surgery on a Trapped-ion Quantum Computer – https://arxiv.org/pdf/2404.16728

[50] Hong, et. al. – Entangling four logical qubits beyond break-even in a nonlocal code – https://arxiv.org/pdf/2406.02666

[51] Steinacker, et. al – A 300mm foundry silicon spin qubit unit cell exceeding 99% fidelity in all operations – https://arxiv.org/pdf/2410.15590

[52]  Michielis, et. al – Silicon spin qubits from laboratory to industry – https://www.researchgate.net/publication/371059405_Silicon_spin_qubits_from_laboratory_to_industry

[53] Hu, et. al – Single-Electron Spin Qubits in Silicon for Quantum Computing – https://www.researchgate.net/publication/390026535_Single-Electron_Spin_Qubits_in_Silicon_for_Quantum_Computing

[54] Meunier, et. al – Silicon spin qubits: a viable path towards industrial manufacturing of large-scale quantum processors – https://link.springer.com/article/10.1140/epja/s10050-025-01514-8

[55] Takeda, et. al – Quantum error correction with silicon spin qubits – https://www.nature.com/articles/s41586-022-04986-6

[56] Wikipedia – Nitrogen-vacancy center – https://en.wikipedia.org/wiki/Nitrogen-vacancy_center

[57] Zhang, et. al – Scalable universal quantum gates between nitrogen-vacancy centers in levitated nanodiamonds arrays – https://arxiv.org/html/2504.08194v1

[58] Ladd, et. al – Quantum Computing – https://ar5iv.labs.arxiv.org/html/1009.2267

[59] Zhong, et. al – Quantum computational advantage using photons – https://arxiv.org/pdf/2012.01625

[60] Nourbakhsh, et. al – QUANTUM COMPUTING: FUNDAMENTALS, TRENDS AND PERSPECTIVES FOR CHEMICAL AND BIOCHEMICAL ENGINEERS – https://arxiv.org/pdf/2201.02823

[61] Weber, et. al – Quantum computing with defects’ (Nitrogen-vacancy) – https://arxiv.org/pdf/1003.1754

[62] Aguado, et. al – Majorana qubits for topological quantum computing – https://pubs.aip.org/physicstoday/article/73/6/44/909657/Majorana-qubits-for-topological-quantum

[63] Bruzewicz, et. al – Trapped-Ion Quantum Computing: Progress and Challenges – https://arxiv.org/pdf/1904.04178

[64] Kadowaki, et al. – Quantum Annealing in the Transverse Ising Model – https://arxiv.org/pdf/cond-mat/9804280

[65] Kirkpatrick, et. al – Optimization by Simulated Annealing – https://www2.stat.duke.edu/~scs/Courses/Stat376/Papers/TemperAnneal/KirkpatrickAnnealScience1983.pdf

[66] Johnson, et. al – Quantum annealing with manufactured spins – https://convexoptimization.com/TOOLS/manufacturedspins.pdf

[67] Amin, et. al – Quantum Boltzmann Machine – https://arxiv.org/pdf/1601.02036

[68] D-Wave – Performance gains in the D-Wave Advantage2 system at the 4,400-qubit scale – https://www.dwavequantum.com/media/wakjcpsf/adv2_4400q_whitepaper-1.pdf

[69] D-Wave – Theoretical Characterization of Adiabatic Quantum Computing and Quantum Annealing – https://www.dwavequantum.com/media/hzjg3lav/14-1082a-a_theoretical_characterization_of_aqc_and_qa.pdf

[70] Zhang, et. al – Cyclic quantum annealing: searching for deep low-energy states in 5000-qubit spin glass – https://www.nature.com/articles/s41598-024-80761-z

[71] Feynman, R. – Quantum Mechanical Computers – https://opn-web-afd-d3bfbkd5bcc5asbs.z02.azurefd.net/opn/media/images/pdfs/11557/11557_23417_110730.pdf?t=638452577230103415

[72] Vert, et. al – Benchmarking quantum annealing with maximum cardinality matching problems – https://www.frontiersin.org/journals/computer-science/articles/10.3389/fcomp.2024.1286057/full

[73] Ding, et. al – Experimenting with D-Wave quantum annealers on prime factorization problems – https://www.frontiersin.org/journals/computer-science/articles/10.3389/fcomp.2024.1335369/full

[74] Tayur & Tenneti – Quantum annealing research at CMU: algorithms, hardware, applications – https://www.frontiersin.org/journals/computer-science/articles/10.3389/fcomp.2023.1286860/full

[75] Salmenperä & Nurminen – Software techniques for training restricted Boltzmann machines on size-constrained quantum annealing hardware – https://www.frontiersin.org/journals/computer-science/articles/10.3389/fcomp.2023.1286591/full

[76] Chern, et. al – Tutorial: calibration refinement in quantum annealing – https://www.frontiersin.org/journals/computer-science/articles/10.3389/fcomp.2023.1238988/full#B8

[77] Pelofske, et. al – Short-depth QAOA circuits and quantum annealing on higher-order ising models – https://www.nature.com/articles/s41534-024-00825-w

[78] King, et. al – Computational supremacy in quantum simulation – https://arxiv.org/abs/2403.00910

[79] Bauza & Lidar – Scaling Advantage in Approximate Optimization with Quantum Annealing – https://arxiv.org/abs/2401.07184

[80] Amin, et. al – Quantum error mitigation in quantum annealing – https://arxiv.org/pdf/2311.01306

[81] King, et. al – Quantum critical dynamics in a 5000-qubit programmable spin glass – https://arxiv.org/pdf/2207.13800

[82] Tasseff, et. al – On the Emerging Potential of Quantum Annealing Hardware for Combinatorial Optimization – https://arxiv.org/abs/2210.04291

[83] Pelofske, et. al – Parallel quantum annealing – https://www.nature.com/articles/s41598-022-08394-8

[84] McGeoch, C. – Theory versus practice in annealing-based quantum computing – https://www.sciencedirect.com/science/article/pii/S0304397520300529?via%3Dihub

[85] Aharonov, et. al – Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation – https://arxiv.org/pdf/quant-ph/0405098

[86] Albash & Lidar – Adiabatic Quantum Computing – https://arxiv.org/pdf/1611.04471

[87] Farhi, et. al – Quantum Computation by Adiabatic Evolution – https://arxiv.org/pdf/quant-ph/0001106

[88] Pakin, S. – Targeting Classical Code to a Quantum Annealer – https://dl.acm.org/doi/pdf/10.1145/3297858.3304071

[89] Lanting, et. al – Entanglement in a quantum annealing processor – https://arxiv.org/pdf/1401.3500

[90] Sarandy & Lidar – Adiabatic quantum computation in open systems – https://arxiv.org/pdf/quant-ph/0502014

[91] Venegas-Andraca, et. al – A cross-disciplinary introduction to quantum annealing-based algorithms – https://arxiv.org/pdf/1803.03372

[92] Vinci & Lidar – Non-stoquastic Hamiltonians in quantum annealing via geometric phases – https://arxiv.org/pdf/1701.07494

[93] Michielsen, et. al – Benchmarking gate-based quantum computers – https://pdf.sciencedirectassets.com/271575/1-s2.0-S0010465517X0009X …

[94] Preskill, J. – QUANTUM COMPUTING AND THE ENTANGLEMENT FRONTIER – https://arxiv.org/pdf/1203.5813

[95] Kockum & Nori – Quantum bits with Josephson junctions – https://arxiv.org/pdf/1908.09558

[96] Martinis & Osborne – Superconducting Qubits and the Physics of Josephson Junctions – https://web.physics.ucsb.edu/~martinisgroup/classnotes/finland/LesHouchesJunctionPhysics.pdf

[97] Quantum News – What is a Josephson Junction? – https://quantumzeitgeist.com/what-is-a-josephson-junction/

[98] Chen, et. al – Current induced hidden states in Josephson junctions – https://www.nature.com/articles/s41467-024-52271-z

[99] Bardin, et. al – Microwaves in Quantum Computing – https://tf.nist.gov/general/pdf/3119.pdf

[100] Wikipedia – Charge qubit – https://en.wikipedia.org/wiki/Charge_qubit#:~:text=In%20quantum%20computing%2C%20a%20charge,Cooper%20pairs%20in%20the%20island)

[101] Farrington, M. – A Brief Introduction to Superconducting Charge Qubits – https://homes.psd.uchicago.edu/~sethi/Teaching/P243-W2021/Final%20Papers/MFarrington_Advanced_Quantum_Mechanics_Final_Project%20(1).pdf

[102] Devoret, et. al – ‘​Superconducting Qubits: A Short Review – https://www.researchgate.net/publication/1875651_Superconducting_Qubits_A_Short_Review

[103] Gonzalez-Meza, et. al – Charge sensitivity in the transmon regime – https://arxiv.org/html/2508.03973v1

[104] Wikipedia – Flux qubit – https://en.wikipedia.org/wiki/Flux_qubit#:~:text=Flux%20qubits%20are%20fabricated%20using,normally%20aluminum%20oxide)%20is%20deposited

[105] Yan, et. al – The flux qubit revisited to enhance coherence and reproducibility – https://www.nature.com/articles/ncomms12964

[106] christianb93 – Josephson Junctions – https://github.com/christianb93/QuantumComputing/blob/master/Implementations/JosephsonJunctions.pdf

[107] Kim, et. al – Superconducting flux qubit with ferromagnetic Josephson π-junction operating at zero magnetic field – https://www.nature.com/articles/s43246-024-00659-1

[108] Chang, et. al – Reproducibility and control of superconducting flux qubits – https://arxiv.org/pdf/2207.01427

[109] Rastelli, et. al – Coherent dynamics in long fluxonium qubits – https://iopscience.iop.org/article/10.1088/1367-2630/17/5/053026

[110] Moskalenko, et. al – High fidelity two-qubit gates on fluxoniums using a tunable coupler – https://www.nature.com/articles/s41534-022-00644-x

[111]  Sharifi, J. – Superconductor qubits hamiltonian approximations effect on quantum state evolution and control – https://www.researchgate.net/publication/352485854_Superconductor_qubits_hamiltonian_approximations_effect_on_quantum_state_evolution_and_control

[112] Wikipedia – Phase qubit – https://en.wikipedia.org/wiki/Phase_qubit

[113] Martinis, et. al – Superconducting Phase Qubits – https://web.physics.ucsb.edu/~martinisgroup/papers/Martinis2008.pdf

[114] Purmessur, et. al – Operation of a high-frequency, phase-slip qubit – https://arxiv.org/html/2502.07043v1

[115] Wikipedia – Superconducting quantum computing – https://en.wikipedia.org/wiki/Superconducting_quantum_computing

[116] Place, et. al – New material platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds – https://www.nature.com/articles/s41467-021-22030-5#:~:text=Abstract,fidelities%20in%20multi%2Dqubit%20processors.

[117]  Ivezic, M. – Transmon Qubits 101 – https://postquantum.com/quantum-computing/transmon-qubits-101/#:~:text=as%20Rigetti%20Computing).-,Key%20Characteristics,large%20capacitor%20shunting%20the%20junction.

[118] Wikipedia – Transmon – https://en.wikipedia.org/wiki/Transmon

[119] Roth, et. al – An Introduction to the Transmon Qubit for Electromagnetic Engineers – https://arxiv.org/pdf/2106.11352

[120] Kang, et. al – New design of three-qubit system with three transmons and a single fixed-frequency resonator coupler – https://www.nature.com/articles/s41598-025-94448-6

[121] Tuokkola, et. al – Methods to achieve near-millisecond energy relaxation and dephasing times for a superconducting transmon qubit – https://www.nature.com/articles/s41467-025-61126-0

[122] Hida, et. al – Flux-Trapping Fluxonium Qubit – https://arxiv.org/html/2505.02416v1#:~:text=the%20gate%20speed.-,Report%20issue%20for%20preceding%20element,13%2C%2014%2C%2015%5D%20.

[123] Wang, et. al – High-coherence fluxonium qubits manufactured with a wafer-scale-uniformity process – https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.23.044064

[124] Randeria, et. al – Dephasing in Fluxonium Qubits from Coherent Quantum Phase Slips – https://arxiv.org/pdf/2404.02989

[125] Nongthombam, et. al – A fluxonium qubit-based hybrid electromechanical system – https://arxiv.org/html/2508.17105v1#:~:text=The%20fluxonium%20qubit%2C%20formed%20by,8%2C%209%2C%2010%5D%20.

[126] Rower, et. al – Suppressing Counter-Rotating Errors for Fast Single-Qubit Gates with Fluxonium – https://journals.aps.org/prxquantum/pdf/10.1103/PRXQuantum.5.040342

[127] Koch, et. al – Charge insensitive qubit design derived from the Cooper pair box – https://arxiv.org/pdf/cond-mat/0703002

[128] Houck, et. al – Life after charge noise: recent results with transmon qubits – https://arxiv.org/pdf/0812.1865

[129] Zhao, et. al – A microwave-activated high-fidelity three-qubit gate scheme for fixed-frequency superconducting qubits – https://arxiv.org/pdf/2504.21346

[130] Bal, et. al – Systematic improvements in transmon qubit coherence enabled by niobium surface encapsulation – https://www.nature.com/articles/s41534-024-00840-x.epdf?sharing_token=qW2WIbkEvgn8OQxbpMYDudRgN0jAjWel9jnR3ZoTv0PNin_LnD5xCZC4sIOFTImvjxS1M-BfTvtP9oIgJ7arA-MDrJ7YDkAappOpPIfu9qQERsyqztWnaEcl8hrzH1xwEJwgSpbFqMUSsjAUSaH5QKAk2XBWbw_QZ_VOuoGctyo%3D

[131] Li, et. al – Superfluid-Cooled Transmon Qubits under Optical Excitation – https://journals.aps.org/prxquantum/pdf/10.1103/q99s-lrnv

[132] Bravyi, et. al – High-threshold and low-overhead fault-tolerant quantum memory – https://arxiv.org/pdf/2308.07915

[133] Google Quantum AI – Quantum error correction below the surface code threshold – https://www.nature.com/articles/s41586-024-08449-y

[134] Gao, et. al – Establishing a New Benchmark in Quantum Computational Advantage with 105-qubit Zuchongzhi 3.0 Processor – https://arxiv.org/pdf/2412.11924

[135] Jiang, et. al – Advancements in superconducting quantum computing – https://academic.oup.com/nsr/article/12/8/nwaf246/8165689

[136] Arnold, et. al – All-optical single-shot readout of a superconducting qubit – https://arxiv.org/pdf/2310.16817

[137] Warner, et. al – Coherent control of a superconducting qubit using light – https://arxiv.org/pdf/2310.16155

[138] Van Damme, et. al – Advanced CMOS manufacturing of superconducting qubits on 300 mm wafers – https://www.nature.com/articles/s41586-024-07941-9

[139] Gayatri & Veni – Material-Driven Optimization of Transmon Qubits for Scalable and Efficient Quantum – https://arxiv.org/pdf/2508.05339

[140] Versluis, et. al – Scalable quantum circuit and control for a superconducting surface code – https://arxiv.org/pdf/1612.08208

[141] Hutchings, et. al – Tunable superconducting qubits with flux-independent coherence – https://arxiv.org/pdf/1702.02253

[142] Strand, et. al – First-order sideband transitions with flux-driven asymmetric transmon qubits – https://arxiv.org/abs/1301.0535

[143] Mooij, et. al – Josephson Persistent-Current Qubit – https://pure.rug.nl/ws/files/3156486/1999ScienceMooij.pdf

[144] You, et. al – Low-decoherence flux qubit – https://arxiv.org/abs/cond-mat/0609225

[145] Yan, et. al – The flux qubit revisited to enhance coherence and reproducibility – https://www.nature.com/articles/ncomms12964

[146] Brooks, et. al – Protected gates for superconducting qubits – https://arxiv.org/pdf/1302.4122

[147] Gyenis, et. al – Experimental Realization of a Protected Superconducting Circuit Derived from the 0–𝜋 Qubit – https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.2.010339

[148] Groszkowski, et. al – Coherence properties of the 0-πqubit – https://arxiv.org/pdf/1708.02886

[149] Houck, et. al – Controlling the spontaneous emission of a superconducting transmon qubit – https://arxiv.org/pdf/0803.4490

[150] Schreier, et. al – Suppressing Charge Noise Decoherence in Superconducting Charge Qubits – https://arxiv.org/pdf/0712.3581

[151] Manucharyan, et. al – Fluxonium: single Cooper pair circuit free of charge offsets – https://arxiv.org/pdf/0906.0831

[152] Nguyen, et. al – The high-coherence fluxonium qubit – https://arxiv.org/pdf/1810.11006

[153] Zhang, et al. – High-fidelity superconducting quantum processors via laser-annealing of transmon qubits – https://arxiv.org/pdf/2012.08475

[154] Wang, et. al – Towards practical quantum computers: transmon qubit with a lifetime approaching 0.5 milliseconds – https://www.nature.com/articles/s41534-021-00510-2

[155] Leghtas, et. al – Hardware-efficient autonomous quantum error correction – https://arxiv.org/pdf/1207.0679

[156] Mirrahimi, et. al. – Dynamically protected cat-qubits: a new paradigm for universal quantum computation – https://arxiv.org/pdf/1312.2017

[157] Guillaud & Mirrahimi – Repetition Cat Qubits for Fault-Tolerant Quantum Computation – https://arxiv.org/abs/1904.09474

[158] Putterman, et. al – Preserving phase coherence and linearity in cat qubits with exponential bit-flip suppression – https://arxiv.org/html/2409.17556v1

[159] Putterman, et. al – Hardware-efficient quantum error correction via concatenated bosonic qubits – https://arxiv.org/pdf/2409.13025

[160] Ruiz, et. al – LDPC-cat codes for low-overhead quantum computing in 2D – https://pmc.ncbi.nlm.nih.gov/articles/PMC11762751/pdf/41467_2025_Article_56298.pdf

[161] Brooks, et. al – Protected gates for superconducting qubits – https://arxiv.org/pdf/1302.4122

[162] Groszkowski, et. al – Coherence properties of the 0-π qubit – https://arxiv.org/abs/1708.02886

[163]  Gyenis, et. al – Experimental realization of an intrinsically error-protected superconducting qubit – https://arxiv.org/pdf/1910.07542

[164] Kolesnikow, et. al – Protected phase gate for the 0-π qubit using its internal modes – https://arxiv.org/pdf/2503.14634

[165] Colwell, B. – The Complete History Of Superconducting Qubits: From Josephson Junctions To Fault-Tolerant Quantum Computing – https://briandcolwell.com/the-complete-history-of-superconducting-qubits-from-josephson-junctions-to-fault-tolerant-quantum-computing/