Superconducting Quantum Computing: A Review

• Quantum computers promise efficient solutions for problems that are intractable for classical computers, including factoring and quantum simulation.Source: §I · p. 1
• Superconducting qubits have rapidly improved in scale and quality, including 99.4% two-qubit gate fidelity in 2014 and a 53-qubit quantum-supremacy demonstration in 2019.Source: §I · p. 1
• The review covers superconducting-qubit theory, design, control, readout, error correction, quantum algorithms, experimental progress, and future perspectives.Source: §I · p. 1
• Superconducting systems offer designable Hamiltonians, semiconductor-compatible fabrication, relatively easy coupling, and microwave-compatible control and measurement.Source: B. Advantages of superconducting qubits · p. 2
• The main hardware challenges are short coherence times, unwanted higher-level transitions, and the need for larger-capacity dilution refrigerators.Source: B. Advantages of superconducting qubits · p. 2
• Charge, flux, and phase qubits use different circuit variables and bias controls to define their relevant quantum states and energy structures.Source: Figure 1 · p. 2

C. Three superconducting qubit archetypes

• The three archetypes are charge, flux, and phase qubits, distinguished by the ratio EJ/EC.Charge qubits have EJ ≪ EC, flux qubits have EJ/EC much greater than 1 but below 100, and phase qubits have EJ ≫ EC.Source: C. Three superconducting qubit… · p. 2

1. Transmon-type qubit

• Transmon reduces charge sensitivity by increasing EJ/EC with a large parallel capacitor while retaining sufficient anharmonicity.The large capacitor lowers the charge energy and produces EJ/EC∼100.Source: §1 · pp. 2–3, Figure 2 · p. 3
• Xmon modifies Transmon with a cross capacitor, resonant-cavity coupling, and independent XY and Z control lines.Its qubits can also be directly coupled through capacitance.Source: §1 · p. 3, Figure 3 · p. 3
• Gmon uses a tunable inductive junction to control coupling strength, avoiding fixed-coupling frequency crowding but adding decoherent channels and layout complexity.A generic tunable-coupler design instead uses a flux-tunable center mode in a three-body chain and does not introduce additional components.Source: §1 · p. 3, Figure 4 · p. 3, Figure 5 · p. 3
• 3D Transmon replaces planar cavities with a three-dimensional waveguide cavity, reducing surface dielectric sensitivity and providing a controlled electromagnetic environment.Scalability remains its major difficulty for large-scale devices.Source: §1 · pp. 3–4, Figure 6 · p. 4

2. 3-JJ flux qubit

• The 3-JJ flux qubit uses a micrometer-sized loop whose opposite persistent-current directions encode the two qubit states.Pulsed microwave modulation of the enclosed magnetic flux can create superpositions of these states.Source: §2 · p. 4

3. C-shunt flux qubit

• C-shunt flux qubits add a capacitor across the smaller junction to reduce charging energy and suppress charge-noise effects.The architecture is described as suppressing both flux and charge noise.Source: §3 · p. 4, Figure 8 · p. 4

Fluxonium

• Fluxonium connects a series array of large-capacitance junctions in parallel with a small junction, making the array behave as an inductive wire below the plasma frequency.The architecture was proposed to address inductance and offset-charge-noise problems.Source: §4 · p. 4, Figure 9 · p. 4

0-π qubit

• The 0-π qubit uses a symmetrical circuit with an interleaved double-well potential, localizing its ground states in separate 0 and π valleys.The corresponding ground-state transition matrix elements are very small.Source: §5 · p. 5, Figure 10 · p. 5

Hybrid qubit

• A hybrid system couples superconducting flux qubits to diamond NV centers to combine their distinct properties.Flux qubits provide control, while NV centers provide longer coherence times.Source: §6 · p. 5, Figure 11 · p. 5

E. Qubit’s lifetime

• Improved qubit structures, parameters, preparation techniques, and materials have greatly enhanced superconducting-qubit lifetime.The review summarizes this development in a table covering the highest reported value for each year.Source: E. Qubit’s lifetime · p. 5, Table I · p. 5

F. Number of entangled qubits

• 2019 marked the first demonstration of quantum supremacy using 53 superconducting qubits.The number of entangled superconducting qubits is presented as a critical indicator of full multi-qubit control.Source: F. Number of entangled qubits · p. 5

1. XY operating principle

• Single-qubit rotations use microwave control, with the drive phase determining the rotation axis in the XY-plane when the qubit is resonant.The drive is capacitively coupled to an Xmon qubit and is expressed as Ω(t) = Ωx cos(ωdt − φ).Source: §1 · p. 6

1. Frequency tuning gates

• A SQUID loop changes the qubit frequency by generating extra magnetic flux when current is applied through its control line.The loop consists of two Josephson junctions and is integrated beneath the Xmon.Source: §2 · p. 6, Figure 13 · p. 6
• The rotation angle around the X-axis can be controlled by changing only the magnitude and duration of the current.This provides a direct current-based control mechanism for the rotation.Source: §2 · p. 6

3. Virtual Z operation

• Virtual Z operations apply a phase offset to subsequent X and Y microwave gates without consuming experimental time.Only the phases in the microwave pulse sequence are modified.Source: §3 · p. 6
• Any single-qubit gate can be decomposed as U(θ, φ, λ) = Zφ · Xθ · Zλ for suitable angles.The decomposition combines Z and X rotations into a general single-qubit operation.Source: §3 · p. 6
• Two-qubit gates require a coupling Hamiltonian, commonly involving σxσx and σyσy terms in superconducting systems.For resonant Xmon qubits, the coupling produces iSWAP and sqrt(iSWAP) gates at selected interaction times.Source: B. Two-qubit gates · pp. 6–7

1. Frequency tuning gates

• Fast adiabatic tuning realizes a CZ gate by bringing |1, 1⟩ near the avoided crossing with |0, 2⟩, producing a selective π phase on |1, 1⟩.Barends et al. reported 99.4% fidelity for an adiabatically tuned two-qubit gate.Source: §1 · p. 7, Figure 14 · p. 7
• 40 ns gate times and 99.54 ± 0.08% fidelity were achieved for a non-adiabatic CZ gate.Diabatic iSWAP-like and CPHASE gates were also reported with fidelities of 0.9966(2) and 0.9954(2), respectively, in 18 ns.Source: §1 · p. 7

2. Cross resonance

• Cross-resonance gates entangle fixed-frequency qubits by applying a microwave drive to two coupled qubits.Interleaved randomized benchmarking reported fidelities exceeding 99% with a 160 ns gate time.Source: §2 · p. 7, Figure 15 · p. 8

3. Parametric gates

• Parametric gates modulate flux bias to drive population between |1, 1⟩ and |0, 2⟩, accumulating a geometric phase for CZ operations.Reported CZ fidelities reached 98.8%, while an eight-qubit processor achieved 93% average process fidelity for three two-qubit gates.Source: §3 · pp. 7–8

4. Resonator-induced gates

• Resonator-induced phase gates use a shared driven bus to implement CZ gates between qubit pairs while returning the cavity to vacuum.In a four-qubit 3D cQED system, fidelities ranged from 96.55% to 98.53% across 12 qubit pairs.Source: §4 · p. 8, Figure 16 · p. 8

D. Multi-qubit gates

• Multi-qubit gates have been demonstrated with fidelities of 86.8% for CCZ, 93.3% for non-adiabatic CCZ, and 81.7% for CCCZ.These results were obtained in superconducting circuits using optimized waveforms or quantum process tomography.Source: D. Multi-qubit gates · p. 8

Dispersive readout

• Dispersive readout detects the qubit state by measuring the transmission coefficient of a coupled readout resonator.The qubit and resonator are coupled capacitively or inductively.Source: A. Dispersive readout · p. 9
• When the qubit-resonator detuning greatly exceeds the coupling strength, the qubit shifts the resonator frequency by the dispersive shift χ.This frequency change is reflected in the measured transmission coefficient.Source: A. Dispersive readout · p. 9
• The measurement integrates the outgoing microwave signal into an I+iQ point, separating |0⟩ and |1⟩ into distinguishable clusters.The resonator response therefore provides a state-dependent signal in the complex plane.Source: A. Dispersive readout · p. 9

High-fidelity single-shot readout

• Dispersive readout alone is often insufficient for fast, high-fidelity single-shot performance.Purcell filters and parametric amplifiers are typically added to improve the measurement.Source: B. High-fidelity single-shot readout · p. 9

Purcell filters

• The Purcell effect limits qubit lifetime through resonator-mediated photon emission, so filters suppress emission near the qubit frequency while remaining transparent at the readout frequency.The lifetime is inversely proportional to qubit-resonator coupling strength and readout-resonator bandwidth.Source: §1 · p. 9
• A quarter-wave bandpass Purcell filter enabled simultaneous measurement of four qubits with 99% intrinsic fidelity in less than 200 ns.The design targets multiplexed fast measurement without increasing environmental damping.Source: §1 · pp. 9–10
• A stepped-impedance Purcell filter improved qubit T1 by up to a factor of 9 relative to predicted unfiltered devices.Its wide stopband protects qubits across a broad frequency range.Source: §1 · p. 10
• 98.25% readout fidelity in 48 ns and 99.2% in 88 ns were achieved using optimized circuit parameters, a Purcell filter, and phase-sensitive parametric amplification.These results combine filtering and amplification to improve fast readout.Source: §1 · p. 10

Parametric amplifiers

• Josephson parametric amplifiers reduce noise through nearly nondissipative pump-to-signal conversion, but traditional JPAs generally have bandwidths no greater than 30 MHz.This bandwidth and dynamic-range limitation constrains scaling to large systems.Source: §2 · p. 10
• Broadband amplifiers use engineered transmission lines or thousands of Josephson junctions to increase amplification bandwidth and gain.IMPA changes characteristic impedance along a tapered line, while TWPA uses phase matching in a junction array.Source: §3 · p. 10, §4 · p. 10

Real-time quantum feedback

• Real-time quantum feedback can support quantum error correction and maintenance of quantum coherence.Its full loop must combine readout, data analysis, and feedback generation before decoherence occurs.Source: C. Real-time quantum feedback · p. 11

Quantum error correction

• Quantum error correction remains necessary for large-scale computation despite progress in coherence time, gate fidelity, and readout fidelity.The review therefore surveys several experimentally demonstrated correction schemes.Source: §V · p. 11

Repetition code

• Repetition codes protect against either bit-flip or phase-flip errors, but cannot detect both error types simultaneously.They are one-dimensional variants of the surface code.Source: A. surface code · p. 11

Surface code

• Surface-code experiments progressed from repetition-code demonstrations toward two-dimensional parity measurements and planar-lattice error detection.The two-dimensional surface code can detect both bit-flip and phase-flip errors, but requires advanced feedback and repeated detection techniques.Source: A. surface code · pp. 11–12

Bosonic codes

• Bosonic codes encode information in a multidimensional oscillator space rather than duplicating many two-level qubits.Cat and binomial codes are examples of this hardware-efficient approach.Source: B. Bosonic codes · p. 12
• Experiments demonstrated universal gates on cat-code logical qubits and ancilla-enabled operations designed to limit coherence loss from ancilla errors.These operations address the need to manipulate encoded logical qubits for universal error correction.Source: B. Bosonic codes · p. 12

Other error-correction methods

• The code was implemented in a five-qubit transmon device, and logical-space randomized benchmarking reduced two-qubit infidelity from 5.8(2)% to 0.60(3)%.The reported reduction was approximately one order of magnitude.Source: C. Other error-correction method · p. 13

Scope

• Current quantum devices remain small-scale and have not surpassed small demonstration algorithms.The review presents these demonstrations as useful for understanding system performance and architecture.Source: §VI · p. 13

Quantum simulation

• Superconducting transmon processors support analog simulation when their Hamiltonians resemble the Bose-Hubbard model.Experiments resolved the entire energy spectrum of a Hamiltonian on a linear transmon array.Source: A. Quantum simulation · p. 13
• Digital quantum simulation uses gate decompositions and has been applied to quantum Rabi, spin, fermionic, and anyonic models.Unlike analog simulation, it is not constrained by the types of systems that can be simulated.Source: A. Quantum simulation · pp. 13–14

Quantum algorithms

• A four-qubit superconducting processor implemented the HHL algorithm for solving a two-dimensional system of linear equations.The review also describes homomorphic encryption for linear-equation solving on a cloud quantum computer.Source: B. Quantum algorithm demonstration · p. 14
• VQE experiments calculated the H2 energy spectrum with near chemical accuracy and addressed molecular problems up to BeH2 using six qubits.Parameterized hybrid algorithms are presented as promising for NISQ devices.Source: B. Quantum algorithm demonstration · pp. 14–15
• Superconducting processors have implemented variational classifiers, quantum kernel estimation, and quantum generative models for machine-learning tasks.These demonstrations present routes to quantum machine learning on NISQ hardware.Source: B. Quantum algorithm demonstration · p. 15

• Quantum supremacy means demonstrating that a programmable quantum device can solve a problem that classical computers practically cannot.Source: C. Quantum supremacy · p. 15, Figure 35 · p. 16
• Google’s 2019 experiment used a programmable superconducting processor with 53 available qubits called Sycamore.Source: C. Quantum supremacy · pp. 15–16, Figure 37 · p. 16
• The experiment implemented a depth-20 two-qubit circuit containing 430 two-qubit and 1,113 single-qubit gates, with predicted total fidelity of 0.2%.Source: C. Quantum supremacy · p. 16
• Improved two-qubit gate fidelity and reduced crosstalk during parallel gate operations were critical techniques in the demonstration.Source: C. Quantum supremacy · p. 16
• The review identifies qubit connectivity, gate fidelity, and coherence time as key challenges, with useful applications and error-correcting quantum computers still needed.Source: §VII · p. 16