Source-linked AI summary
Validating quantum computers using randomized model circuits
Andrew W. Cross, Lev S. Bishop, Sarah Sheldon, Paul D. Nation, Jay M. Gambetta
TL;DR
Two-qubit circuits can contain remote interactions that require hardware-aware mapping and optimization. The paper describes circuit passes and approximate two-qubit-gate expansions, obtaining high fidelities and reducing SWAP overhead and effective infidelity.
Problem
Two-qubit circuits may contain gates acting on remote qubit pairs, creating a need for circuit mapping and SWAP-gate reduction.
Method
The approach combines layer-by-layer SWAP mapping, contiguous two-qubit block collection, Local Ordering Circuit Optimization, and mirrored approximate gate expansions.
Results
0.997 median fidelity is achieved by the 2-basis gate approximations, while mirroring reduces mean basis-gate applications from 2.2 to 2.0 for F_b = 0.97.
Takeaways & Limitations
Approximate expansions can improve effective gate performance, with mirroring reducing infidelity by a factor of 0.74 at F_b = 0.97.
Abstract
from arXiv · showhide
We introduce a single-number metric, quantum volume, that can be measured using a concrete protocol on near-term quantum computers of modest size ($n\lesssim 50$), and measure it on several state-of-the-art transmon devices, finding values as high as 16. The quantum volume is linked to system error rates, and is empirically reduced by uncontrolled interactions within the system. It quantifies the largest random circuit of equal width and depth that the computer successfully implements. Quantum computing systems with high-fidelity operations, high connectivity, large calibrated gate sets, and circuit rewriting toolchains are expected to have higher quantum volumes. The quantum volume is a pragmatic way to measure and compare progress toward improved system-wide gate error rates for near-term quantum computation and error-correction experiments.
Appendix A: Qiskit transpiler passes
The transpiler rewrites model circuits into the target system’s gate set and connectivity while applying passes that reduce implementation overhead and depth.
- Connectivity handling: The transpiler represents available two-qubit couplings with a connectivity graph whose vertices are qubits and edges are executable interactions.Directed edges encode the permitted coupling direction for particular systems.
- Compilation pipeline: Model circuits are expanded into the IBM Q gate set, including CNOT and single-qubit gates, before system-specific rewriting.The unrolling pass descends through hierarchical gate definitions and replaces them with lower-level gates.
- Gate rewriting: CNOT reorientation uses an identity when only the reverse directed edge is available, whereas the pass fails when neither direction is supported.This adapts CNOT direction to the target connectivity graph.
- Gate rewriting: The CNOT cancellation pass removes repeated identical CNOTs in pairs and retains one gate when the repetition count is odd.Sequences with even m are removed, while odd m becomes a single CNOT.
- Gate rewriting: Single-qubit optimization replaces each same-qubit sequence with at most one gate, targeting fewer physical pulses.The pass composes gates using rewriting rules and selects among u1, u2, and u3 implementations.
- Connectivity handling: The swap mapping pass converts layers containing remote two-qubit gates into nearest-neighbor circuits while allowing a qubit permutation and minimizing depth.It greedily selects swap circuits that bring coupled qubits closer together.
7. Single-qubit optimization pass
The optimized compiling workflow applies analysis and rewriting passes that collect two-qubit blocks, resynthesize them, and reorder qubits to reduce connectivity overhead.
- 2. Two-qubit block optimization pass: The two-qubit block collection pass traverses gates topologically and gathers the largest previously unseen contiguous blocks on CNOT control and target qubits.It returns disjoint blocks for subsequent optimization.
- 2. Two-qubit block optimization pass: Each collected block is replaced by a newly synthesized sub-circuit using either exact KAK decomposition or approximate synthesis.The replacement computes the block unitary before resynthesis.
- 2. Two-qubit block optimization pass: LOCO permutes qubits so CNOT-connected pairs are as near-neighbor as possible for a linear nearest-neighbor topology.It uses weighted reverse Cuthill-McKee ordering to minimize the sparse interaction matrix bandwidth.
Appendix B: Approximate compiling
The appendix develops approximate decompositions of two-qubit target unitaries using a fixed basis gate, optionally exploiting mirrored gates, and evaluates their fidelity and gate-count benefits.
- Two-qubit unitaries are represented using canonical information-content coordinates (α, β, γ) restricted to the Weyl chamber, with local-unitary equivalence preserving those coordinates.
- A fixed basis gate requires zero, one, or multiple applications depending on the target unitary; three applications of a CNOT-equivalent gate suffice for exact expansion.
- Mirroring selects between a target gate and its SWAP-composed counterpart, improving the available approximation fidelity and potentially reducing inserted SWAP gates.
- 0.997 is the reported median average fidelity for two-basis-gate approximations, comparable to typical two-qubit gate fidelities in current devices.
- At basis fidelity Fb = 0.97, mirroring changes the mean basis-gate count from 2.2 to 2.0 and yields effective fidelities Fe = 0.976 and F(m)e = 0.978.
- For Fb = 0.97, approximate expansions reduce effective infidelity by factors of 0.82 without mirroring and 0.74 with mirroring.
Appendix C: Confidence intervals for the heavy probability
The appendix sets a stricter empirical threshold above 2/3 and uses finite-sample confidence analysis to decide whether a model-circuit width and depth succeed.
- Success requires the estimated heavy fraction ˆhd to exceed a threshold t greater than 2/3, testing whether the true heavy probability hd exceeds 2/3.
- The estimate combines nc random model circuits, ns executions per circuit, and the total heavy-outcome count nh.
- The conservative analysis assumes each circuit’s conditional heavy probability is either zero or one, making nh/ns binomially distributed with parameter nc.
- A minimum of nc = 100 circuits is required for the normal approximation used near the 2/3 threshold.
- For nc = 5000 circuits, claiming success requires an observed heavy fraction above t = 0.68 using a one-sided 2-sigma confidence bound.
Appendix D: Device parameters
The device-parameter appendix identifies the four IBM Q systems, the selected experimental qubits and connectivity, and the error-rate quantities averaged for those qubits.
- The experiments used 5-qubit Tenerife, 16-qubit Melbourne, and 20-qubit Tokyo and Johannesburg devices.
- The listed error rates were measured one day before the quantum-volume experiments.
- Table V reports average single-qubit error ϵ1Q, two-qubit error ϵCX, and measurement error ϵM for the experimental qubits.
- The four qubits used for each m = d = 4 experiment are highlighted in the device diagrams, with CX gates available along highlighted connections.