Quantum simulation and computing with Rydberg-interacting qubits

M. Morgado; S. Whitlock

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Quantum simulation and computing with Rydberg-interacting qubits

M. Morgado, S. Whitlock

arXiv:2011.03031v2Nov 5, 2020quant-phcond-mat.quant-gasphysics.atom-ph

TL;DR

Rydberg-interacting atom arrays address the challenge of building programmable quantum simulators and computers at useful scale. This review synthesizes their qubit encodings, operations, interactions, simulations, and computing schemes, reporting demonstrations beyond 100 qubits and progress toward larger reliable circuits. The platform's flexibility and strong interactions support a transition toward general-purpose quantum applications, although substantial challenges remain.

Problem
Quantum processors must scale programmable control and interaction capabilities to study quantum dynamics and solve problems difficult for classical computers.
Method
The review surveys Rydberg qubit encodings, mediated interactions, quantum gates, many-body Hamiltonians, simulations, computing schemes, and remaining challenges.
Results
More than 100 interacting qubits, high-fidelity entangling operations with F > 0.991, native multiqubit gates, and highly entangled states involving ∼20 qubits have been demonstrated.
Takeaways & Limitations
Rydberg systems are transitioning from few- and many-body experiments toward more general-purpose quantum simulation and computing applications.

Abstract

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Arrays of optically trapped atoms excited to Rydberg states have recently emerged as a competitive physical platform for quantum simulation and computing, where high-fidelity state preparation and readout, quantum logic gates and controlled quantum dynamics of more than 100 qubits have all been demonstrated. These systems are now approaching the point where reliable quantum computations with hundreds of qubits and realistically thousands of multiqubit gates with low error rates should be within reach for the first time. In this article we give an overview of the Rydberg quantum toolbox, emphasizing the high degree of flexibility for encoding qubits, performing quantum operations and engineering quantum many-body Hamiltonians. We then review the state-of-the-art concerning high-fidelity quantum operations and logic gates as well as quantum simulations in many-body regimes. Finally, we discuss computing schemes that are particularly suited to the Rydberg platform and some of the remaining challenges on the road to general purpose quantum simulators and quantum computers.

I. INTRODUCTION:

Quantum simulation and computing use controlled interactions among quantum objects to access dynamics and problems difficult for classical devices. Rydberg platforms combine programmable atomic arrays, strong interactions, and flexible control, with recent demonstrations advancing toward larger, deeper quantum processors.

Quantum simulation prepares quantum objects in defined states and transforms them through controlled interactions to study otherwise classically difficult dynamics.
Analog simulators reproduce a target Hamiltonian, whereas digital simulators encode states in qubit registers and apply programmable gate sequences.
Rydberg systems use programmable optical-tweezer arrays, individually addressable atoms, and strong, tunable interactions for analog and digital operations.
Arrays with on the order of 100 deterministically filled sites, high-fidelity readout, native multiqubit gates, and controlled dynamics beyond 100 qubits have been demonstrated.Entangling operations reached F > 0.991, and GHZ states involving ∼20 qubits were generated.
The review covers qubit encodings, Rydberg-mediated interactions, logic gates, many-body Hamiltonians, technological progress, and Rydberg-suited computing schemes.
The largest reliably executable random-circuit depth is defined as D□ = ⌊ϵ−1/2⌋, while current demonstrations are compatible with D□≈10.Assuming realistic technological improvements, several hundred qubits with D□≳40 are anticipated.
Parallel readout of entire arrays is routinely achieved within ≳10 ms, with best reported fidelity > 0.996.

B. Types of Rydberg qubits

Rydberg qubits are classified by how ground and Rydberg states encode information, with each class offering different interaction, control, and coherence trade-offs. Ground-Rydberg qubits favor easy manipulation and fast entangling operations, while Rydberg-Rydberg encodings provide flexible interactions but harder local addressing.

Rydberg qubits are classified into three main types according to the number of ground and Rydberg states forming the qubit.The types differ in characteristic energy scales and whether optical or microwave fields manipulate the states.
Ground-Rydberg (gr) qubits: Ground-Rydberg qubits encode |0⟩ in a weakly interacting state and |1⟩ in a strongly interacting Rydberg state.
Ground-Rydberg (gr) qubits: Ground-Rydberg interactions are generally always on, complicating isolated single-qubit operations, but these qubits are relatively easy to initialize, manipulate, and measure.They support fast ≲100 ns and high-fidelity entangling operations, as well as many-body state engineering and simulation.
Ground-Rydberg (gr) qubits: Collective ground-Rydberg qubits can simplify assembly of larger arrays and enhance atom-light coupling, but atom-number fluctuations and short-range interactions limit their present use.
Rydberg-Rydberg (rr) qubits: Rydberg-Rydberg qubits encode both logical states in different Rydberg states, enabling flexible interactions including longer-range dipolar exchange.Their typical energy splittings are in the 10−60 GHz range and scale approximately with n−3.
Rydberg-Rydberg (rr) qubits: Rydberg-Rydberg qubits are typically manipulated with microwave fields, reducing Doppler dephasing but making individual addressing more difficult.
Rydberg-Rydberg (rr) qubits: Alkaline-earth Rydberg atoms showed T∗2 = 22 µs, while small alkali arrays maintained coherent dynamics for approximately ∼(4−7) µs with 1−2.5 MHz nearest-neighbor interactions.Circular Rydberg states in cryogenic environments may extend lifetimes but substantially increase technical complexity.

Ground-ground (gg) qubits

Ground-ground qubits use long-lived low-lying atomic states, combining strong coherence with switchable interactions mediated through Rydberg excitation. They support high-fidelity single-qubit control and demonstrated two-qubit gates, while motional and scattering effects remain relevant control considerations.

Qubit properties: Ground-ground qubits encode information in two long-lived low-lying atomic states, offering long coherence times and switchable interactions.Implementations include hyperfine ground-state pairs or an electronic ground state paired with a metastable excited state.
Qubit properties: Seconds-range dephasing times are achievable with spin echo or magic trapping, compared with typical unrefocused T∗_2 = (1−20) ms.The longer coherence is identified as a significant advantage over ground-Rydberg and Rydberg-Rydberg encodings.
Control and gates: F > 0.9999 single-qubit fidelity has been demonstrated for global manipulation, while local addressing reached F = 0.992 with low crosstalk.These results distinguish global control from individually addressed operations.
Control and gates: Two-qubit gates between ground-ground qubits have reached F = 0.974 and F = 0.891 with individual qubit addressing.Because ground-ground qubits interact weakly, gates mediate interactions by timed or shaped excitation to and de-excitation from Rydberg states.
Atom-light control: Two-photon optical Raman transitions drive ground-ground qubits, using large intermediate-state detuning to suppress photon scattering.The coupling is produced by phase-coherent fields, while off-resonant scattering contributes decay and dephasing.

Global versus local manipulation

Rydberg platforms support both global and local manipulation, including low-crosstalk single- and two-qubit operations. However, fully independent control currently extends to only a small number of qubits at once, motivating sequential or parallel operation patterns.

Local manipulation: Focused Rydberg or Raman lasers have demonstrated single- and two-qubit operations with negligible neighboring-qubit influence.Local AC Stark shifts combined with global microwave couplings also enable high-fidelity, low-crosstalk addressing, including in 3D registers.
Control limits: Fully independent manipulation has not yet reached much more than two qubits at a time.This remains a stated technological boundary for Rydberg-qubit control.
Global versus local control: Larger-system demonstrations therefore primarily use sequential one- or two-qubit operations or parallel operations applied similarly across multiple qubits.Global fields remain sufficient for some quantum simulations and Hamiltonian-evolution computing schemes.
Single-qubit gates: For constant Ω and ∆ over gate time τg, the single-qubit evolution is represented as a unitary rotation.The rotation angle is ˜Ωτg, around an axis determined by Ω, ∆, and ϕ.
Single-qubit gates: With ∆ = 0, xy rotations generate a complete set of single-qubit rotations, including Rx, Ry, and Rz constructions.Focused-laser AC Stark shifts can implement native Rz rotations, and these operations combine with multiqubit gates into a universal set.

Rydberg-Rydberg interactions

Rydberg interactions arise from large transition dipole moments and can mediate gates or engineer many-body Hamiltonians. Their effective form depends on Förster defect, producing resonant dipolar exchange or off-resonant van der Waals interactions.

Interaction origin: Rydberg-Rydberg interactions originate from large transition dipole moments and can mediate quantum gates or engineer many-body Hamiltonians.The interaction strength depends on interatomic separation and the selected Rydberg states.
Interaction regimes: |∆F| > |µacµbd|/R^3 produces a van der Waals energy shift with 1/R^6 distance dependence.This regime commonly occurs for atoms prepared in identical or not directly dipole-coupled Rydberg states.
Interaction model: A few dominant dipole-coupled pair states often provide a first approximation to the interaction model.The selected states may belong to one species or represent different computational and auxiliary qubit types.
Interaction model: Förster defect ∆F determines the effective interaction regime after far-detuned pair states are neglected.The two-atom Hamiltonian is written in the pair-state basis {|ab⟩, |cd⟩}, with Vj,k as the distance-dependent interaction coefficient.
Interaction regimes: ∆F ≈ 0 yields coherent dipolar state exchange with 1/R^3 distance dependence.The exchange couples |ab⟩ and |cd⟩ through the interaction and its Hermitian conjugate.
Interaction control: External electric fields can shift pair states into Förster resonance and thereby modulate effective two-body interaction strength.Such modulation may benefit certain gate protocols.

Native two-qubit gates

Rydberg qubits support diverse native two-qubit gates, using blockade, phase accumulation, collective oscillations, and dipolar exchange. These protocols offer flexible control but can involve always-on interactions outside the addressed pair.

Protocol overview: Rydberg two-qubit protocols use computational and auxiliary states, with π- and 2π-pulses arranged sequentially or in parallel.The protocols cover gr- and gg-qubits; blockade and Rydberg-interacting transitions mediate the operations.
Controlled rotations: A CUxy gate performs a target-qubit Uxy rotation conditioned on the control state and can become a CNOT for θ = π and ϕ = 0 with a preceding control Rz(−π/2) rotation.Strong interactions block the target oscillation when the control is in |1⟩c.
Robustness and constraints: CUxy gates are robust to interaction-strength variation, atomic motion, and mechanical forces, but always-on computational-subspace interactions can make them evolve other qubits in larger systems.The gate can also extend to k control qubits within the blockade volume.
Collective rotations: A parallel CUxy variant drives collective oscillations between zero and one shared Rydberg excitation and produces a maximally entangled Bell state for θ = π.The protocol addresses both qubits in parallel, so it has no control-target distinction.
Controlled phase gates: A CPHASE gate uses transient excitation to an auxiliary Rydberg state, with interaction-dependent phase accumulation distinguishing computational basis states.The canonical CZ case has Φ00 = Φ10 = Φ01 = 0 and Φ11 = π.
Exchange gates: A four-pulse native XY protocol exploits resonant dipolar exchange without individual addressing and is predicted to achieve F > 0.991.Its 1/R3 interactions may support quantum-information routing and digital spin-model simulation.

Multi-qubit (more than two-qubit) Rydberg gates

Rydberg interactions enable native gates acting on more than two qubits, including multi-control, fan-out, and generalized phase gates. These constructions can reduce gate overhead, although some implementations require asymmetric interactions or complex pulse optimization.

Toffoli gates: A three-qubit Toffoli gate can require six two-qubit CNOT gates and nine single-qubit gates when decomposed, motivating native Rydberg implementations.The figure reports an experimentally demonstrated C2NOT fidelity of F > 0.870(4) corrected for readout.
Controlled phase gates: The native CkZ blockade gate applies a target π phase only when all k controls begin in |1⟩, with otherwise-blocked target rotation.Asymmetric control-target interactions avoid unwanted interactions between control qubits.
Experimental protocols: A parallel C2Z gate has been demonstrated using a globally amplitude- and frequency-modulated pulse whose complex shape was found by numerical optimal-control optimization.The approach is analogous to a parallel CZ implementation but requires a more complex pulse shape.
Generalizations: Proposed extensions include multiple targets, Deutsch gates, short-pulse Toffoli protocols, dark-state adiabatic evolution, and resonant-exchange-based CkNOT and CNOTk gates.Some proposals generalize the C2Z gate or adapt the CUxy protocol to single-step k-qubit Toffoli operations.
Multi-qubit gate families: CkNOT gates condition a target flip on multiple controls, while CNOTk gates condition multiple target flips on one control.Native multiqubit gates can be advantageous because decomposed implementations require many one- and two-qubit operations.

E. Many-body Hamiltonians

Rydberg qubits naturally realize many-body Hamiltonians, with experiments so far focusing mainly on quantum spin-1/2 models. These systems support studies of quantum dynamics, phase transitions, and controlled state preparation.

Experimental focus: Experimental demonstrations involving more than a few interacting Rydberg qubits have focused mainly on quantum spin-1/2 Hamiltonians.These models are central to condensed-matter and nonequilibrium physics.
Scientific applications: Rydberg many-body systems are used to study quantum dynamics, quantum phase transitions, and quasi-adiabatic or non-adiabatic state-preparation protocols.The reviewed regimes involve interacting systems beyond only a few qubits.

Quantum Ising model (gr- and gg-qubits)

Rydberg-interacting qubits naturally realize transverse- and longitudinal-field Ising models, whose tunable interactions produce crystalline phases and blockade-constrained PXP dynamics. Related Rydberg encodings also implement XY spin-exchange models and the SSH model.

Quantum Ising model: The quantum Ising model uses laser-controlled transverse and longitudinal fields together with diagonal, finite-range Rydberg interactions.The interaction strength and range depend on the selected Rydberg states and atomic separations.
Quantum Ising model: Multiple crystalline ground-state phases emerge when |Ω/V| ≲1, with phase structure determined by the detuning-to-interaction ratio ∆/V.In one dimension, phases are labeled by broken translational symmetry, such as Zq=2 for |0101010...⟩.
Quantum Ising model: Floating crystal phases can surround the ordered phases, while two-dimensional square and kagome lattices exhibit competing ordered phases and exotic phase transitions.The phase boundaries near Ω→0 are determined by detunings involving the Riemann zeta function.
Quantum Ising model: The blockade radius Rb controls whether interactions strongly constrain the Hilbert space or remain weak relative to the lattice spacing.For Rb≫a, nearby simultaneous excitations are prevented and the effective Hilbert space is substantially reduced.
PXP model: Nearest-neighbor blockade with a<Rb<2a yields an effective PXP model in which Rabi rotations occur only when neighboring qubits are in |0⟩.Its constrained configuration graph has Fibonacci-size growth and includes special non-ergodic eigenstates associated with quantum scars.
XY and SSH models: Dipolar exchange between s- and p-state rr-qubits realizes an XY model, which has been used to simulate an SSH chain with alternating weak and strong links.The exchange interaction scales as C3(θj,k)/R^3 and can be supplemented by microwave-driven single-qubit rotations.

XXZ and other spin models (rr-qubits)

Rydberg qubits encoded in different excited states provide direct routes to XXZ and XY spin models, while experiments demonstrate entangling operations ranging from Bell-state generation to high-fidelity optical-clock-qubit gates.

XXZ model: Two Rydberg states coupled through a common intermediate state realize an anisotropic XXZ model with state-dependent interaction strength and anisotropy.The interaction strength scales as Jj,k=−C6/R^6.
XXZ model: The XXZ platform has been studied in disordered three-dimensional gases and describes interactions between qubits encoded in circular Rydberg states.Related spin models with spin-nonconserving terms have also been proposed and studied.
Entangling operations: Bell-state entanglement is characterized through parity oscillations after a global Uxy rotation, with FBell derived from populations and oscillation contrast.States with FBell>0.5 are identified as entangled.
Entangling operations: τg=1.85 µs was much shorter than the measured coherence time T∗2=10(1) ms, in principle allowing thousands of operations if fidelities improve.The comparison concerns ground-state encoded qubits.
Entangling operations: F>0.991(4) Bell-state fidelity was demonstrated for 88Sr gr-qubits using pCUxy(π) entangling operations.The operation used a 51 ns gate time in a dimerized one-dimensional array.
Entangling operations: Fast 51 ns gates combined with high fidelity were reported as encouraging for deep circuits with many qubits, with D□≳10.The work targeted integration of Rydberg gates with ultra-coherent optical clock qubits.

B. Gate implementations and elementary quantum circuits

Rydberg platforms support controlled gates between heterogeneous qubits, multi-qubit operations, and site-specific control in larger arrays. Demonstrations include heteronuclear gates, ion-mediated gates, and addressing of 121-site neutral-atom arrays.

Heterogeneous gates: A heteronuclear Rydberg gate between 87Rb and 85Rb atoms achieved raw fidelity F=0.73(1) and Bell-state fidelity FBell=0.59(3).The reported limitations were power fluctuations in the Rydberg excitation lasers and Doppler dephasing.
Heterogeneous gates: Heteronuclear gates can support architectures using distinct computational and ancilla species.The passage presents this as a possibility enabled by two-species Rydberg operations.
Ion-mediated gates: Rydberg-mediated CPHASE gates between 88Sr+ ions achieved FBell=0.78(3) and accelerated operations by several orders of magnitude.The gate was limited mainly by microwave power fluctuations and finite Rydberg-laser coherence.
Scalable addressing: A 2D array of 121 sites demonstrated site-specific single- and two-qubit operations using focused 3 µm Rydberg-excitation beams and crossed AODs.The array had an average filling of 0.55 and used local AC-Stark-shifted microwave pulses.
Multi-qubit gates: A no-individual-addressing pCZ protocol produced a Bell-state raw fidelity of 0.959(2) in a 1D 87Rb array.The protocol was characterized using parity oscillations.
Multi-qubit gates: Three-qubit C2Z and Toffoli gates were demonstrated in trimerized arrays with asymmetric nearest-neighbor blockade.Their pulse sequences were more complicated than the corresponding two-qubit pCZ protocol.

C. Many-body quantum state engineering

Rydberg systems engineer many-body states through controlled parameter ramps, quenches, and tailored interactions, producing magnetic order, crystalline states, GHZ states, and symmetry-protected topological phases. Scaling is constrained by decay, dephasing, and the difficulty of quantifying many-body coherence.

Ising-state preparation: Controlled ramps of ∆(t) and Ω(t) prepare antiferromagnetically ordered Ising states, with correlations measured through spin-spin correlation functions.Experiments used different spatial configurations and lattice geometries.
Limitations: Ramp speed and correlation strength are limited by Rydberg decay and ground-Rydberg dephasing, while relevant many-body coherence may differ from T2 or TRabi.The difference depends on the simulated model, noise spectrum, and many-body effects such as mechanical forces.
Crystalline states: Slow detuning sweeps prepared Z2 crystalline states with fidelity 0.77(6) for N=7 and 0.009(2) for N=51.The constrained Hilbert-space dimension scales approximately as ([1+√5]/2)^N under blockade.
Nonequilibrium dynamics: Quenching from a crystalline state produced unusually long-lived oscillations later associated with weak ergodicity breaking and many-body scars.Related experiments studied quantum Kibble-Zurek dynamics and periodically driven revivals.
Ising-state preparation: Improved excitation lasers increased measured coherence lengths in 2D arrays beyond 100 qubits to 7 sites and more than 11 sites.These experiments demonstrated controlled adiabatic evolution near the limits of state-of-the-art numerical methods.
GHZ-state preparation: For N=20, computational-basis measurements showed large peaks at the two antiferromagnetically ordered configurations, while parity oscillations characterized GHZ entanglement.The corresponding figure reports a lower-bound fidelity F≥0.54(2).
Topological states: A 14-qubit dipolar-interaction chain prepared a half-filled SSH ground state whose robustness to perturbations is characteristic of a symmetry-protected topological phase.The system exhibited coherent dynamics up to approximately 6 µs.

IV. TOWARD MORE PROGRAMMABLE QUANTUM SIMULATIONS AND QUANTUM COMPUTATIONS

The section reviews Rydberg-suited computational models, including quantum annealing through the LHZ architecture and quantum cellular automata. These proposals exploit local Rydberg interactions and programmable control to address optimization, entanglement generation, and other quantum-information tasks.

A. Rydberg quantum annealing: Quantum annealing encodes optimization problems in the ground state of an Ising spin Hamiltonian and obtains solutions by measuring the final qubit states.The Hamiltonian uses local fields and two-body interactions, while an adiabatic transformation connects an easily prepared initial ground state to the target Hamiltonian.
A. Rydberg quantum annealing: All-to-all interactions required by standard quantum annealing are not native to Rydberg systems, motivating the LHZ mapping to local interactions.The mapping introduces physical qubits representing relative orientations of logical spins and requires additional four-body constraints.
A. Rydberg quantum annealing: A Rydberg-dressed LHZ annealer numerically prepares the ground state with success probability > 75% within ∼0.5 ms for 4 all-to-all connected logical spins.The demonstrated instance uses 8 physical qubits and 3 ancillas, with success approaching unity for slower sweeps.
B. Quantum cellular automata: Quantum cellular automata use homogeneous local interactions that are periodic in space and time, replacing classical bits and update rules with qubits and multiqubit operations.The model supports parallel quantum evolution without local addressing and can exhibit reversibility, superposition, measurement probabilities, and entanglement.
B. Quantum cellular automata: Multifrequency Rydberg excitation can implement totalistic QCA rules and extend them with dissipative conditional interactions through an auxiliary state.The rules can operate continuously in parallel or discretely through block partitioning.
B. Quantum cellular automata: Numerical simulations of systems up to 9 qubits demonstrate feasible engineering of highly entangled states using unitary, continuous-time, and non-unitary QCA rules.Purely unitary discrete-time generation of a GHZ_N state requires at least (N −1)/2 timesteps.

C. Variational quantum algorithms

The section presents variational quantum algorithms as hybrid quantum-classical methods and reviews their proposed use with Rydberg blockade interactions. It also situates these approaches alongside scalability, coherence, circuit-depth, and readout challenges for Rydberg processors.

C. Variational quantum algorithms: Variational quantum algorithms use quantum processors to generate trial wavefunctions while classical optimization adjusts parameters based on measured expectation values.The approach targets ground-state preparation for molecular Hamiltonians, combinatorial optimization, and metrologically useful states.
C. Variational quantum algorithms: QAOA avoids quasi-adiabatic evolution and may tolerate some noise, but its applicability to general optimization problems remains relatively uncertain beyond depth d = 1.The passage identifies this as a scope limitation rather than a demonstrated failure.
C. Variational quantum algorithms: Rydberg blockade provides a natural implementation of QAOA for the unit-disk maximum independent set problem.The problem seeks the largest set of mutually unconnected vertices in a two-dimensional plane.
C. Variational quantum algorithms: QAOA performs similarly to quantum annealing for small systems, requiring 10^2−10^3 measurements when the annealing time is constrained to T = 10/Ω.The comparison concerns the required measurement count under the stated time constraint.
C. Variational quantum algorithms: Realistic-parameter estimates indicate that near-term Rydberg processors could address computationally hard problems with N ∼10^2−10^3 vertices.A related analysis finds Rydberg QAOA relatively noise resilient under spontaneous emission, with further improvement possible through objective-function choice.
V. Future challenges and opportunities: Rydberg processors face additional scaling and hardware constraints from particle loss, coherence limits, destructive readout, and finite circuit depth.The review discusses deterministic loading, laser and motional dephasing, atom lifetimes, parallel gates, and quantum non-demolition readout as relevant considerations.

Conclusion

The conclusion presents Rydberg-mediated neutral-atom arrays as a versatile platform that has progressed from few- and many-body experiments toward general-purpose quantum simulation and computing. Continued technological and algorithmic advances are projected to support larger circuits and applications across science and optimization.

Conclusion: Rydberg platforms now combine programmable arrays of ∼100 qubits, high-fidelity entangling operations, diverse atomic species, specialized gates, large-scale entanglement, and programmable simulations with ≳100 interacting qubits.The conclusion lists these as breakthrough achievements since the previous major review.
Conclusion: These achievements mark a transition toward general-purpose quantum simulation and quantum computing applications.The transition is described as having just begun.
Conclusion: Continuing improvements are expected to enable ∼1000-qubit circuits with effective circuit depths D□≳40.The review places this projected regime beyond the current state of the art in any platform.
Conclusion: Projected applications include many-body physics, quantum state engineering, quantum chemistry, materials science, mathematics, optimization, and machine learning.The conclusion also anticipates hybrid quantum-classical computing and other undiscovered applications.

Appendix A: Achievable circuit depth D□

The appendix defines effective circuit depth D□ as the largest square circuit that can be reliably executed before an error is likely. This benchmark incorporates device errors and implementation overheads and can be measured directly on hardware.

Appendix A: Achievable circuit depth D□: Effective circuit depth D□ compares quantum devices by finding the largest square circuit with n = d that remains reliable before an error occurs.The circuits apply parallel two-qubit gates or pairwise-interaction Hamiltonian evolution to all qubits.
Appendix A: Achievable circuit depth D□: The model circuit applies local two-qubit operations in parallel, with the red square marking the reliable region and the yellow cross marking the first error.Dark shaded rectangles indicate the causal cone through which quantum correlations spread.
Appendix A: Achievable circuit depth D□: D□ corresponds to Ng = D□^2/2 gates inside the square circuit, with average per-gate error probability ϵ = 1/(2Ng).This formulation captures different error channels and overheads associated with physical gate implementation.
Appendix A: Achievable circuit depth D□: Assuming system-size-independent error per gate, reliable digital circuits satisfy nd × ϵ ≤1, giving D□=⌊ϵ−1/2⌋.For digital circuits, ϵ can be approximated by 1 −F, where F is the two-qubit-operation fidelity.
Appendix A: Achievable circuit depth D□: Effective circuit depth can also be determined directly with randomized circuits and statistical tests, providing a hardware benchmark analogous to quantum volume.The appendix notes that quantum volume scales as VQ ∼2D□ under the cited convention.