Minimal universal two-qubit controlled-NOT-based circuits
Open accessAbstract
We give quantum circuits that simulate an arbitrary two-qubit unitary operator up to a global phase. For several quantum gate libraries we prove that gate counts are optimal in the worst and average cases. Our lower and upper bounds compare favorably to previously published results. Temporary storage is not used because it tends to be expensive in physical implementations. For each gate library, the best gate counts can be achieved by a single universal circuit. To compute the gate parameters in universal circuits, we use only closed-form algebraic expressions, and in particular do not rely on matrix exponentials. Our algorithm has been coded in $\mathrm{C}++$.
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
Clever partitioning of circuits can achieve good parallelism and efficient use of distributed memory resources for particular supercomputer architectures.
“Clever partitioning of circuits can achieve good parallelism and efficient use of distributed memory resources for particular supercomputer architectures [21-27].”
From Validating quantum computers using randomized model circuits (2019) · cites this paper as [22]
For circuits defined on a 2-dimensional square grid of qubits, the state-of-the-art classical simulation is d = 40 for a 9 × 9 grid.
“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].”
From Validating quantum computers using randomized model circuits (2019) · cites this paper as [22]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Cites
<i>Quantum Computation and Quantum Information</i>
Nielsen, M. A., Chuang, I., & Grover, L. K. (2002) · American Journal of Physics · doi:10.1119/1.1463744 · Cited by 21,895 · see citing works
Quantum computation and quantum information
Law, J. (2001) · ACM SIGSOFT Software Engineering Notes · doi:10.1145/505482.505499 · Cited by 20,449 · see citing works
Mixed-state entanglement and quantum error correction
Bennett, C. H., DiVincenzo, D. P., Smolin, J. A., & Wootters, W. K. (1996) · Physical Review A · doi:10.1103/physreva.54.3824 · Cited by 5,272 · see citing works
Elementary gates for quantum computation
Barenco, A., Bennett, C. H., Cleve, R., DiVincenzo, D. P., Margolus, N., Shor, P., Sleator, T., Smolin, J. A., & Weinfurter, H. (1995) · Physical Review A · doi:10.1103/physreva.52.3457 · Cited by 4,311 · see citing works
Entanglement of a pair of quantum bits
Hill, S., & Wootters, W. K. (1997) · Physical Review Letters · doi:10.1103/physrevlett.78.5022 · Cited by 2,823 · see citing works
Quantum dynamics of single trapped ions
Leibfried, D., Blatt, R., Monroe, C., & Wineland, D. (2003) · Reviews of Modern Physics · doi:10.1103/revmodphys.75.281 · Cited by 2,739 · see citing works
Two-bit gates are universal for quantum computation
DiVincenzo, D. P. (1995) · Physical Review A · doi:10.1103/physreva.51.1015 · Cited by 1,069 · see citing works
Time optimal control in spin systems
Khaneja, N., Brockett, R. W., & Glaser, S. J. (2001) · Physical Review A · doi:10.1103/physreva.63.032308 · Cited by 755 · see citing works
Guillemin, V., & Pollack, A. (2010) · American Mathematical Society eBooks · doi:10.1090/chel/370 · Cited by 667 · see citing works
Optimal quantum circuits for general two-qubit gates
Vatan, F., & Williams, C. (2004) · Physical Review A · doi:10.1103/physreva.69.032315 · Cited by 443 · see citing works
Strong-Driving-Assisted Multipartite Entanglement in Cavity QED
Solano, E., Agarwal, G. S., & Walther, H. (2003) · Physical Review Letters · doi:10.1103/physrevlett.90.027903 · Cited by 403 · see citing works
Makhlin, Y. (2002) · Quantum Information Processing · doi:10.1023/a:1022144002391 · Cited by 354 · see citing works
Universal quantum circuit for two-qubit transformations with three controlled-NOT gates
Vidal, G., & Dawson, C. M. (2004) · Physical Review A · doi:10.1103/physreva.69.010301 · Cited by 277 · see citing works
Characterization of separable states and entanglement witnesses
Lewenstein, M., Kraus, B., Horodecki, P., & Cirac, J. I. (2001) · Physical Review A · doi:10.1103/physreva.63.044304 · Cited by 181 · see citing works
Computing local invariants of quantum-bit systems
Grassl, M., Rötteler, M., & Beth, T. (1998) · Physical Review A · doi:10.1103/physreva.58.1833 · Cited by 176 · see citing works
Approximation by Quantum Circuits
Knill, E. (1995) · arXiv · doi:10.48550/arxiv.quant-ph/9508006 · Cited by 136 · see citing works
Millman, R., Guillemin, V., & Pollack, A. S. (1978) · American Mathematical Monthly · doi:10.2307/2321081 · Cited by 135 · see citing works
Polynomial invariants of quantum codes
Rains, E. M. (2000) · IEEE Transactions on Information Theory · doi:10.1109/18.817508 · Cited by 95 · see citing works
Exact Two-Qubit Universal Quantum Circuit
Zhang, J., Vala, J., Sastry, S., & Whaley, K. B. (2003) · Physical Review Letters · doi:10.1103/physrevlett.91.027903 · Cited by 68 · see citing works
Arbitrary two-qubit computation in 23 elementary gates
Bullock, S. S., & Markov, I. L. (2003) · Physical Review A · doi:10.1103/physreva.68.012318 · Cited by 59 · see citing works
Self-citationCanonical decompositions of n-qubit quantum computations and concurrence
Bullock, S. S., & Brennen, G. K. (2004) · Journal of Mathematical Physics · doi:10.1063/1.1723701 · Cited by 43 · see citing works
Self-citationCanonical Decompositions of n-qubit Quantum Computations and Concurrence
Bullock, S. S., & Brennen, G. K. (2003) · CERN Document Server (European Organization for Nuclear Research) · Cited by 32 · see citing works
Self-citationArtin, M. (1993) · Birkhäuser Basel eBooks · doi:10.1007/978-3-0348-9124-0 · Cited by 16 · see citing works
Polynomial invariants of quantum codesPreprint
Rains, E. M. (1997) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/9704042 · Cited by 12 · see citing works
Optimal Realization of an Arbitrary Two-Qubit Quantum GatePreprint
Vatan, F., & Williams, C. P. (2003) · arXiv (Cornell University) · Cited by 9 · see citing works
Flagged first, then by citations
Research works printed but not linked: 3
- PRL 78 , 5022-5025, 1997, quant-ph/9703041 . · arXiv
- [15]M. Grassl et al., 'Computing Local Invariants of Qubit Systems,' PRA 58 , 1833-1839, 1998, quant-ph/9712040 . · arXiv
- [18]D. Wineland, C. Monroe, W. Itano, D. Leibfried, B. King, and D. Meekhof, 'Experimental Issues in Coherent Quantum Manipulation of Trapped Atomic Ions,' NIST Journal of Research 103, p. 259 (1998).
Cited by
Sorted by citations
Validating quantum computers using randomized model circuits
Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D., & Gambetta, J. M. (2019) · Physical Review A · doi:10.1103/physreva.100.032328 · Cited by 704 · see citing works
Quantum Computation of Electronic Transitions Using a Variational Quantum Eigensolver
Parrish, R. M., Hohenstein, E. G., McMahon, P. L., & Martínez, T. J. (2019) · Physical Review Letters · doi:10.1103/physrevlett.122.230401 · Cited by 282 · see citing works
Yao.jl: Extensible, Efficient Framework for Quantum Algorithm Design
Luo, X.-Z., Liu, J.-G., Zhang, P., & Wang, L. (2020) · Quantum · doi:10.22331/q-2020-10-11-341 · Cited by 157 · see citing works
Solving quantum statistical mechanics with variational autoregressive networks and quantum circuits
Liu, J.-G., Mao, L., Zhang, P., & Wang, L. (2021) · Machine Learning: Science and Technology · doi:10.1088/2632-2153/aba19d · Cited by 57 · see citing works
Malinowski, M., Allcock, D. T. C., & Ballance, C. J. (2023) · PRX Quantum · doi:10.1103/prxquantum.4.040313 · Cited by 54 · see citing works
Demonstration of quantum volume 64 on a superconducting quantum computing system
Jurcevic, P., Javadi-Abhari, A., Bishop, L. S., Lauer, I., Bogorin, D. F., Brink, M., Capelluto, L., Günlük, O., Itoko, T., Kanazawa, N., et al. (2021) · Quantum Science and Technology · doi:10.1088/2058-9565/abe519 · Cited by 85 · see citing works
Quinones, M. P., & Junqueira, C. (2020) · arXiv · doi:10.48550/arxiv.2007.13245 · Cited by 6 · see citing works
- Authors
- Shende, V. V., Markov, I. L., & Bullock, S. S.
- Affiliations
- University of Michigan (US)
- National Institute of Standards and Technology (US)
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2004-06-30
- Volume
- 69
- Issue
- 6
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W2046761109
- Cited by
- 288
- References
- 25
- Citations by year
- 2026: 20 · 2025: 29 · 2024: 32 · 2023: 29 · 2022: 25 · 2021: 24 · 2020: 24 · 2019: 7 · 2018: 10 · 2017: 6 · 2016: 4 · 2015: 8 · 2014: 5 · 2013: 10 · 2012: 7
- FWCI
- 6.41
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review A (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Keywords
- CNOT gate, global phase
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
