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Validating quantum computers using randomized model circuits
Open accessAbstract
We introduce a single-number metric, quantum volume, that can be measured using a concrete protocol on near-term quantum computers of modest size ($n\ensuremath{\lesssim}50$), 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.
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TL;DR
The paper introduces quantum volume, a single-number metric measured by random model circuits of equal width and depth using heavy output generation, and reports measurements on IBM Q transmon devices with values as high as 16. It links quantum volume to system error rates, connectivity, and circuit optimization, and provides simulations for how it scales with two-qubit error and coupling map.
Key findings
- Quantum volume is defined as \(V_Q = 2^{\min(d,m)}\), measuring the largest square-shaped model circuit a quantum computer successfully implements; measured values reach as high as 16 (\(\log_2 V_Q = 4\)). On the tested devices, \(\log_2 V_Q = 3\) for Tokyo, \(\log_2 V_Q = 2\) for Tenerife, and \(\log_2 V_Q < 2\) for Melbourne.
- Heavy output probabilities for \(m=d\) circuits include Tokyo \(m=d=2\): 0.718(0.006), \(m=d=3\): 0.682(0.002)*, \(m=d=4\): 0.614(0.003); Johannesburg \(m=d=4\): 0.664(0.004), and optimized \(m=d=4\): 0.699(0.001), against the 2/3 threshold.
- Circuit optimization improves performance: standard Qiskit Terra transpiler produced 28.1 CX gates on average with \(\hat h_d = 0.614(0.003)\); KAK reduced this to 21.0 CX gates with \(\hat h_d = 0.632(0.005)\); approximate SU(4) with 1% CX error used 17.7 CX gates and achieved \(\hat h_d = 0.649(0.005)\). Noisy depolarizing simulations substantially over-estimated experimental performance.
- Connectivity simulations show all-to-all connectivity has an advantage: for \(\log_2 V_Q = 12\), the maximum permissible two-qubit error was 0.0032 for all-to-all, 0.0015 for square grid, and 0.0014 for loop. Measurement error reduces permissible two-qubit error for square grid at \(\log_2 V_Q = 12\) from 0.0015 at 0% to 0.00125 at 1% and 0.0002 at 5%.
Methods
- The protocol uses random model circuits of width \(m\) and depth \(d\), built from layers of random permutations and Haar-random SU(4) two-qubit gates, compiled to the target device’s gate set. Success is evaluated via the heavy output generation problem, requiring heavy output probability \(h_d > 2/3\), with quantum volume defined as \(V_Q = 2^{\min(d,m)}\).
- Experiments were run on four IBM Q devices: 5-qubit Tenerife, 16-qubit Melbourne, 20-qubit Tokyo, and 20-qubit Johannesburg. For each \(m\), 200 circuits were run for \(d=m\) with \(m \in \{2,3,4\}\); repeats used 5000 circuits for \(m=2\) on Tenerife and \(m=3\) on Tokyo, and 1000 circuits for optimized Johannesburg experiments. Results were compared with depolarizing noise simulations.
- Simulations of all-to-all, square-grid, and loop coupling maps up to 12 qubits estimated the two-qubit gate error required for \(\log_2 V_Q = 4, 6, 8, 12\), assuming single-qubit error equal to \(\epsilon/10\) and measurement errors of 0%, 1%, and 5% for the square grid.
Limitations
- The paper states that noisy simulations substantially over-estimate performance, that classical computation of the heavy set scales exponentially with \(m\), and that for error rates as low as \(10^{-4}\) model circuits may still be small enough to simulate classically; lower error rates may require new or modified volume metrics. It also notes that the LOCO optimization did not significantly improve heavy output generation for small numbers of qubits, and that connectivity comparisons involve small total qubit numbers, with asymptotic loop-versus-grid differences potentially larger.
- Evident from the text: experimental validation is limited to four IBM Q transmon devices and small square circuits up to \(m=d=4\) (with \(m=d=5\) on Johannesburg), so larger-device and architecture conclusions rely on simulations. Device error rates were measured one day before the experiments and can fluctuate during the experiment timescale.
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In the paper
“The quantum volume is linked to system error rates, and is empirically reduced by uncontrolled interactions within the system.”
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In the paper
“We note that the noisy simulation substantially over-estimates the performance, highlighting the value of system-level metrics such as quantum volume.”
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In the paper
“the standard Qiskit Terra transpiler [ ? ] produces circuits with 28 CX gates on average, and we measure ˆ h d = 0 . 614(0 . 003); KAK reduces the average number of CX gates to 21 and produces ˆ h d = 0 . 632(0 . 005).”
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In the paper
“the standard Qiskit Terra transpiler [ ? ] produces circuits with 28 CX gates on average, and we measure ˆ h d = 0 . 614(0 . 003); KAK reduces the average number of CX gates to 21 and produces ˆ h d = 0 . 632(0 . 005).”
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In the paper
“the standard Qiskit Terra transpiler [ ? ] produces circuits with 28 CX gates on average, and we measure ˆ h d = 0 . 614(0 . 003); KAK reduces the average number of CX gates to 21 and produces ˆ h d = 0 . 632(0 . 005).”
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In the paper
“The approximate SU(4) circuits introduce further gains with the best result of ˆ h d = 0 . 649(0 . 005) achieved using circuits with a 1% CX error approximation.”
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“For the case m = d = 4 the results lie just below the threshold of ˆ h d = 2 / 3, and optimizing the circuits with both the KAK decomposition and the approximate SU(4) with 1% CX error yields ˆ h d = 0 . 699(0 . 001).”
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“It is clear from Table III that all-to-all connectivity provides an advantage over the less-connected graph; log 2 V Q of 12 is achievable with twice the two-qubit error rate (0.0032) of the square grid (0.0015) and the 12-qubit loop (0.0014).”
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“It is clear from Table III that all-to-all connectivity provides an advantage over the less-connected graph; log 2 V Q of 12 is achievable with twice the two-qubit error rate (0.0032) of the square grid (0.0015) and the 12-qubit loop (0.0014).”
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In the paper
“the standard Qiskit Terra transpiler [ ? ] produces circuits with 28 CX gates on average, and we measure ˆ h d = 0 . 614(0 . 003); KAK reduces the average number of CX gates to 21 and produces ˆ h d = 0 . 632(0 . 005).”
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“This device has the lowest gate error rates of all the devices measured, with single qubit gate errors a factor of four smaller and two qubit gate errors nearly half than those measured on Tokyo .”
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“At the same time, there is little difference between the required two-qubit error rate for the square grid versus the loop graphs; the error rate for the loop is less than 7% lower than that of the square grid for the 12-qubit case.”
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“The 2-basis gate approximations perform surprisingly well, with the median fidelities F (2) avg = 0 . 99, F (2 m ) avg = 0 . 997 comparing favorably to the typical 2-qubit gate fidelities for current quantum devices.”
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“For F b = 0 . 97 we get F e = 0 . 976, F ( m ) e = 0 . 978, reducing the infidelity by factors of 0.82 and 0.74 respectively.”
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“With a fairly noisy basis gate F b = 0 . 97 and no mirroring, the best expansion by this method has 3 applications of the basis for 22%, two applications for 76%, one application for 2%, and zero applications for < 0 . 1% of targets, thus an average of 2.2 basis gate applications.”
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“With the freedom to mirror, three applications for 3%, two applications for 93%, one application for 4%, and zero applications for < 0 . 1% of targets, thus a mean of 2.0 basis gate applications.”
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“Although this optimization did not lead to significant improvements for heavy output generation using small numbers of qubits, we expect SWAP optimizations such as these to further improve results for larger circuits mapped onto devices with limited connectivity.”
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“In particular, we introduced an approximate method for NISQ devices, and used it to show experimental improvements.”
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“We implemented various circuit optimization passes (far from optimal) and showed a measurable change in the experimental performance.”
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“In addition, circuits with a similar form appear in near-term algorithms like quantum adiabatic optimization algorithms [17] and variational quantum eigensolvers [18].”
What the sources say
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“However, random circuits model generic state preparations, and are used as the basis of proposals for demonstrating quantum advantage [16].”
What the sources say
- Not checked yetcites [16]
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In the paper
“Clever partitioning of circuits can achieve good parallelism and efficient use of distributed memory resources for particular supercomputer architectures [21-27].”
What the sources say
- Not checked yetcites [21] Z.-Y. Chen, Q. Zhou, C. Xue, X. Yang, G.-C. Guo, and G.-P. Guo, Science Bulletin 63 , 964 (2018), arXiv: 1802.06952. source
- Not checked yetcites [22] J. Chen, F. Zhang, C. Huang, M. Newman, and Y. Shi, arXiv:1805.01450 [quant-ph] (2018), arXiv: 1805.01450. source
- Not checked yetcites [23] R. Li, B. Wu, M. Ying, X. Sun, and G. Yang, arXiv:1804.04797 [quant-ph] (2018), arXiv: 1804.04797. source
- Not checked yetcites [24] E. Pednault, J. A. Gunnels, G. Nannicini, L. Horesh, T. Magerlein, E. Solomonik, and R. Wisnieff, arXiv:1710.05867 [quant-ph] (2017), arXiv: 1710.05867. source
- Not checked yetcites [25] S. Boixo, S. V. Isakov, V. N. Smelyanskiy, and H. Neven, arXiv:1712.05384 [quant-ph] (2017), arXiv: 1712.05384.
- Not checked yetcites [26] T. H¨ aner, D. S. Steiger, M. Smelyanskiy, and M. Troyer, arXiv:1604.06460 [quant-ph] (2016), arXiv: 1604.06460.
- Not checked yetcites [27] M. Smelyanskiy, N. P. D. Sawaya, and A. Aspuru-Guzik, arXiv:1601.07195 [quant-ph] (2016), arXiv: 1601.07195. source
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“A straightforward wave-vector propagation approach requires exponential space and time ∼ 2 m .”
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In the paper
“This definition differs from [13, 14] and loosely coincides with the complexity of classically simulating the model circuits.”
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“A 'Feynman' algorithm uses linear space ∼ dm but exponential time ∼ 4 dm .”
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“Quantum algorithms are generally not random circuits.”
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“It is possible to trade off time and space complexity in a smooth way [19].”
What the sources say
- Not checked yetcites [19] S. Aaronson and L. Chen, arXiv preprint arXiv:1612.05903 (2016).
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“High-fidelity state preparation and readout are also necessary.”
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“Quantum computing systems with high-fidelity operations, high connectivity, large calibrated gate sets, and circuit rewriting toolchains are expected to have higher quantum volumes.”
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“Particular efforts for circuit partitioning and parallelism have been expended for circuits defined on a 2-dimensional square grid of qubits, where the state-of-the-art is d = 40 for a 9 × 9 grid [22].”
What the sources say
- Not checked yetcites [22] J. Chen, F. Zhang, C. Huang, M. Newman, and Y. Shi, arXiv:1805.01450 [quant-ph] (2018), arXiv: 1805.01450. source
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In the paper
“It is well-known that quantum algorithms can be expressed as polynomial-sized quantum circuits built from two-qubit unitary gates [15].”
What the sources say
- Not checked yetcites [15] source
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In the paper
“While we focus on transmon devices, the protocol can be implemented with any universal programmable quantum computing device.”
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In the paper
“For U b ∼ CNOT ∼ U d ( π/ 4 , 0 , 0) it is well-known [43, 44] that 3 applications of the basis is sufficient to cover all of SU(4).”
What the sources say
- Not checked yetcites [43] F. Vatan and C. Williams, Physical Review A 69 , 032315 (2004).
- Not checked yetcites [44] G. Vidal and C. M. Dawson, Physical Review A 69 , 010301 (2004).
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“Therefore, we heuristically bound the treewidth of the model circuits as min( d, m ), and since the simulation complexity grows exponentially with the treewidth, we define the quantum volume as V Q = 2 min( d,m ) .”
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“we instead require a minimum of n c = 100 circuits and make a normal approximation to the binomial”
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In the paper
“This reordering is efficient, having a runtime that is linear in the number of nonzero matrix elements [39].”
What the sources say
- Not checked yetcites [39] W. M. Chan and A. George, BIT 20 , 8 (1980).
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“Typical trapped-ion systems have both two-qubit gate errors and measurement errors less than 0.01, which based on Table III should be sufficient to achieve log 2 V Q = 6 if not higher.”
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In the paper
“Similarly, we can infer that for superconducting devices, coupling maps with more connectivity should produce higher quantum volume, but only if additional coupling does not also introduce larger errors.”
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In the paper
“Although all operations must improve to increase the quantum volume, the first path is to prioritize improving the gate fidelity above other operations, such as measurement and initialization.”
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In the paper
“To be confident with a finite number of trials that the heavy probability h d exceeds 2 / 3, we should set stricter threshold t > 2 / 3, requiring the estimated heavy probability ˆ h d > t to claim success.”
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In the paper
“This relatively small difference is due to the small total number of qubits, since there is a significant asymptotic difference between loop and grid layouts.”
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In the paper
“This metric takes into account all relevant hardware parameters. This includes the performance parameters (coherence, calibration errors, crosstalk, spectator errors, gate fidelity, measurement fidelity, initialization fidelity) as well as the design parameters such as connectivity and gate set. It also includes the software behind the circuit optimization.”
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“Zhang et al. [45] give decompositions using a more general 'super controlled' basis U b ∼ U d ( π/ 4 , β b , 0), for any β b , both an expansion with 3 applications of U b to decompose an arbitrary U t ∼ U d ( α t , β t , γ t ) and also an expansion using two applications of U b for a restricted target unitary U t ∼ U d ( α t , β t , 0), γ t = 0 for any α t , β t .”
What the sources say
- Not checked yetcites [45] Y.-S. Zhang, M.-Y. Ye, and G.-C. Guo, Physical Review A 71 , 062331 (2005).
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In the paper
“Additionally, the quantum volume is architecture-independent, and can be applied to any system that is capable of running quantum circuits.”
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In the paper
“The IBM Q systems used in this paper accept quantum circuits expressed by products of controlled-NOT (CNOT) gates and single-qubit gates [36].”
What the sources say
- Not checked yetcites [36] A. W. Cross, L. Bishop, J. Smolin, and J. M. Gambetta, arXiv preprint arXiv:1707.03429 (2017).
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The authors state this as an assumption.
The approximation of achievable model circuit depth assumes independent stochastic errors.
“We approximate the achievable model circuit depth ˜ d ( m ) by assuming independent stochastic errors, so that the computation fails with high probability when the model circuit volume (width times depth) satisfies”
The authors state this as an assumption.
In simulations for Table III, the single-qubit gate error is equal to ε/10.
“assuming the single-qubit gate error is equal to ϵ/ 10 (Table III).”
The authors state this as an assumption.
In the Table III estimates, measurements are perfect.
“The estimates are based on simulations using a depolarizing noise model with two-qubit error ϵ as given, single-qubit error equal to ϵ/ 10, and perfect measurements.”
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“Besides, these metrics all involve certain randomness due to synthetically generated inputs or circuits, which do not necessarily share the properties of real quantum workloads.”
From QASMBench: A Low-Level Quantum Benchmark Suite for NISQ Evaluation and Simulation (2023) · cites this paper as [28]
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In the citing paper
“Cross etal. tried to validate the quantum computers using randomized model circuits andhave attempted to introduce a method for NISQ [54].”
From Quantum Computing in the Next-Generation Computational Biology Landscape: From Protein Folding to Molecular Dynamics (2024) · cites this paper as [54]
The cited source says what the claim attributes to it.
In the citing paper
“Regarding QC metrics, Cross et al. deined the quantum volume protocol which quantiied the largest random circuit of equal width and depth that a NISQ device can successfully implement [28], taking gate and measurement error into consideration.”
From QASMBench: A Low-Level Quantum Benchmark Suite for NISQ Evaluation and Simulation (2023) · cites this paper as [28]
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- S. Aaronson, 2017, Proceedings of the 32nd Computational Complexity Conference (CCC '17)
- T. Häner, 2016, Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis (SC '16)
Cited by
Sorted by citations
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Metadata
- Authors
- Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D., & Gambetta, J. M.
- Affiliations
- IBM Research - Thomas J. Watson Research Center (US)
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2019-09-20
- Volume
- 100
- Issue
- 3
- ISSN
- 0556-2791 (ISSN-L)
- 2469-9926
- 2469-9934
- OpenAlex
- W2903516100
- Semantic Scholar
- 3a9f7dda6264e4827e78230d88fea893a9d81b09
- Cited by
- 704
- References
- 44
- Citations by year
- 2026: 77 · 2025: 94 · 2024: 106 · 2023: 115 · 2022: 109 · 2021: 117 · 2020: 69 · 2019: 17
- FWCI
- 35.58
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review A (publishedVersion)
- arXiv (Cornell University) (publishedVersion)
- arXiv (Cornell University)
- Funders
- Army Research Office (W911NF-14-1-0124)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Keywords
- quantum volume, transmon qubits, gate error rates, near-term quantum devices, quantum error correction
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