András Gilyén
ABSTRACT
An n-qubit quantum circuit performs a unitary operation on an exponentially large, 2n-dimensional, Hilbert space, which is a major source of quantum speed-ups. We develop a new “Quantum singular value transformation” algorithm that can directly harness the advantages of exponential dimensionality by applying polynomial transformations to the singular values of a block of a unitary operator. The transformations are realized by quantum circuits with a very simple structure - typically using only a constant number of ancilla qubits - leading to optimal algorithms with appealing constant factors. We show that our framework allows describing many quantum algorithms on a high level, and enables remarkably concise proofs for many prominent quantum algorithms, ranging from optimal Hamiltonian simulation to various quantum machine learning applications. We also devise a new singular vector transformation algorithm, describe how to exponentially improve the complexity of implementing fractional queries to unitaries with a gapped spectrum, and show how to efficiently implement principal component regression. Finally, we also prove a quantum lower bound on spectral transformations.
- Scott Aaronson. 2006. The ten most annoying questions in quantum computing. https://www.scottaaronson.com/blog/?p=112.Google Scholar
- Alexei B. Aleksandrov and Vladimir V. Peller. 2016. Operator Lipschitz functions. Russian Mathematical Surveys 71, 4 (2016), 605–702. arXiv: 1611.01593Google ScholarCross Ref
- Andris Ambainis. 2004.Google Scholar
- Quantum Walk Algorithm for Element Distinctness. In Proceedings of the 45th IEEE Symposium on Foundations of Computer Science (FOCS). 22–31. arXiv: quant-ph/0311001 Google ScholarDigital Library
- Joran van Apeldoorn and András Gilyén. 2019. Improvements in Quantum SDPSolving with Applications. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP). (to appear) arXiv: 1804.05058Google Scholar
- Joran van Apeldoorn and András Gilyén. 2019. Quantum algorithms for zero-sum games. (2019).Google Scholar
- arXiv: 1904.03180Google Scholar
- Joran van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf. 2017.Google Scholar
- Quantum SDP-Solvers: Better upper and lower bounds. In Proceedings of the 58th IEEE Symposium on Foundations of Computer Science (FOCS). 403–414. arXiv: 1705.01843Google Scholar
- Simon Apers and Alain Sarlette. 2018. Quantum Fast-Forwarding Markov Chains. (2018).Google Scholar
- arXiv: 1804.02321Google Scholar
- Sanjeev Arora and Satyen Kale. 2016. A Combinatorial, Primal-Dual Approach to Semidefinite Programs. Journal of the ACM 63, 2 (2016), 12:1–12:35. Earlier version in STOC’07. Google ScholarDigital Library
- Dominic W. Berry, Graeme Ahokas, Richard Cleve, and Barry C. Sanders. 2007.Google Scholar
- Efficient Quantum Algorithms for Simulating Sparse Hamiltonians. Communications in Mathematical Physics 270, 2 (2007), 359–371. arXiv: quant-ph/0508139Google Scholar
- Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma. 2014. Exponential improvement in precision for simulating sparse Hamiltonians. In Proceedings of the 46th ACM Symposium on the Theory of Computing (STOC). 283–292. arXiv: 1312.1414Google Scholar
- Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma. 2015. Simulating Hamiltonian Dynamics with a Truncated Taylor Series. Physical Review Letters 114, 9 (2015), 090502.Google ScholarCross Ref
- arXiv: 1412.4687Google Scholar
- Dominic W. Berry, Andrew M. Childs, and Robin Kothari. 2015.Google Scholar
- Hamiltonian Simulation with Nearly Optimal Dependence on all Parameters. In Proceedings of the 56th IEEE Symposium on Foundations of Computer Science (FOCS). 792–809. arXiv: 1501.01715Google Scholar
- Fernando G. S. L. Brandão, Amir Kalev, Tongyang Li, Cedric Yen-Yu Lin, Krysta M. Svore, and Xiaodi Wu. 2019. Quantum SDP Solvers: Large Speed-ups, Optimality, and Applications to Quantum Learning. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP). (to appear) arXiv: 1710.02581Google Scholar
- Fernando G. S. L. Brandão and Krysta M. Svore. 2017. Quantum Speed-ups for Solving Semidefinite Programs. In Proceedings of the 58th IEEE Symposium on Foundations of Computer Science (FOCS). 415–426. arXiv: 1609.05537Google Scholar
- Gilles Brassard, Peter Høyer, Michele Mosca, and Alain Tapp. 2002. Quantum Amplitude Amplification and Estimation. In Quantum Computation and Quantum Information: A Millennium Volume. Contemporary Mathematics Series, Vol. 305. AMS, 53–74. arXiv: quant-ph/0005055Google Scholar
- Shantanav Chakraborty, András Gilyén, and Stacey Jeffery. 2019.Google Scholar
- The power of block-encoded matrix powers: improved regression techniques via faster Hamiltonian simulation. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP). (to appear) arXiv: 1804.01973Google Scholar
- Andrew M. Childs, Robin Kothari, and Rolando D. Somma. 2017. Quantum Algorithm for Systems of Linear Equations with Exponentially Improved Dependence on Precision. SIAM Journal on Computing 46, 6 (2017), 1920–1950.Google ScholarCross Ref
- arXiv: 1511.02306Google Scholar
- Andrew M. Childs, Dmitri Maslov, Yunseong Nam, Neil J. Ross, and Yuan Su. 2017. Toward the first quantum simulation with quantum speedup. (2017). arXiv: 1711.10980Google Scholar
- Andrew M. Childs and Nathan Wiebe. 2012. Hamiltonian simulation using linear combinations of unitary operations. Quantum Information and Computation 12, 11&12 (2012), 901–924. arXiv: 1202.5822Google Scholar
- Anirban Narayan Chowdhury and Rolando D. Somma. 2017.Google Scholar
- Quantum algorithms for Gibbs sampling and hitting-time estimation. Quantum Information and Computation 17, 1&2 (2017), 41–64. arXiv: 1603.02940 Google ScholarDigital Library
- Yuliya B. Farforovskaya and Ludmila N. Nikolskaya. 2009. Modulus of continuity of operator functions. St. Petersburg Math. J. – Algebra i Analiz 20, 3 (2009), 493–506.Google Scholar
- Richard P. Feynman. 1982.Google Scholar
- Simulating physics with computers. International Journal of Theoretical Physics 21, 6-7 (1982), 467–488.Google Scholar
- Roy Frostig, Cameron Musco, Christopher Musco, and Aaron Sidford. 2016. Principal Component Projection Without Principal Component Analysis. In Proceedings of the 33rd International Conference on Machine Learning (ICML). 2349–2357. Google ScholarDigital Library
- arXiv: 1602.06872Google Scholar
- András Gilyén. 2019.Google Scholar
- Quantum Singular Value Transformation & Its Algorithmic Applications. Ph.D. Dissertation. University of Amsterdam. Advisor(s) Ronald de Wolf.Google Scholar
- András Gilyén and Or Sattath. 2017.Google Scholar
- On Preparing Ground States of Gapped Hamiltonians: An Efficient Quantum Lovász Local Lemma. In Proceedings of the 58th IEEE Symposium on Foundations of Computer Science (FOCS). 439–450. arXiv: 1611.08571Google Scholar
- András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe. 2018. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. {Full version} arXiv: 1806.01838Google Scholar
- Lov K. Grover. 1996. A Fast Quantum Mechanical Algorithm for Database Search. In Proceedings of the 28th ACM Symposium on the Theory of Computing (STOC). 212–219. arXiv: quant-ph/9605043Google ScholarDigital Library
- Lov K. Grover. 2005. Fixed-Point Quantum Search. Physical Review Letters 95, 15 (2005), 150501.Google ScholarCross Ref
- arXiv: quant-ph/0503205Google Scholar
- Jeongwan Haah. 2018. Product Decomposition of Periodic Functions in Quantum Signal Processing. (2018).Google Scholar
- arXiv: 1806.10236Google Scholar
- Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. 2009. Quantum algorithm for linear systems of equations. Physical Review Letters 103, 15 (2009), 150502.Google ScholarCross Ref
- arXiv: 0811.3171Google Scholar
- Aram W. Harrow, Cedric Yen-Yu Lin, and Ashley Montanaro. 2017. Sequential measurements, disturbance and property testing. In Proceedings of the 28th ACMSIAM Symposium on Discrete Algorithms (SODA). 1598–1611. arXiv: 1607.03236 Google ScholarDigital Library
- Yong He, Ming-Xing Luo, E. Zhang, Hong-Ke Wang, and Xiao-Feng Wang. 2017.Google Scholar
- Decompositions of n-qubit Toffoli Gates with Linear Circuit Complexity. International Journal of Theoretical Physics 56, 7 (2017), 2350–2361.Google Scholar
- Peter Høyer. 2000. Arbitrary phases in quantum amplitude amplification. Physical Review A 62, 5 (2000), 052304.Google ScholarCross Ref
- arXiv: quant-ph/0006031Google Scholar
- Camille Jordan. 1875. Essai sur la géométrie à n dimensions. Bulletin de la Société Mathématique de France 3 (1875), 103–174. http://eudml.org/doc/85325Google ScholarCross Ref
- Iordanis Kerenidis and Alessandro Luongo. 2018. Quantum classification of the MNIST dataset via Slow Feature Analysis. (2018).Google Scholar
- arXiv: 1805.08837Google Scholar
- Iordanis Kerenidis and Anupam Prakash. 2017. Quantum gradient descent for linear systems and least squares. (2017).Google Scholar
- arXiv: 1704.04992Google Scholar
- Iordanis Kerenidis and Anupam Prakash. 2017.Google Scholar
- Quantum Recommendation Systems. In Proceedings of the 8th Innovations in Theoretical Computer Science Conference (ITCS). 49:1–49:21. arXiv: 1603.08675Google Scholar
- Shelby Kimmel, Cedric Yen-Yu Lin, Guang Hao Low, Maris Ozols, and Theodore J. Yoder. 2017.Google Scholar
- Hamiltonian simulation with optimal sample complexity. npj Quantum Information 3, 1 (2017), 13. arXiv: 1608.00281Google Scholar
- Seth Lloyd. 1996.Google Scholar
- Universal Quantum Simulators. Science 273, 5278 (1996), 1073–1078.Google Scholar
- Seth Lloyd, Masoud Mohseni, and Patrick Rebentrost. 2014. Quantum principal component analysis. Nature Physics 10 (2014), 631–633. arXiv: 1307.0401Google ScholarCross Ref
- Guang Hao Low and Isaac L. Chuang. 2016. Hamiltonian Simulation by Qubitization. (2016).Google Scholar
- arXiv: 1610.06546Google Scholar
- Guang Hao Low and Isaac L. Chuang. 2017. Hamiltonian Simulation by Uniform Spectral Amplification. (2017).Google Scholar
- arXiv: 1707.05391Google Scholar
- Guang Hao Low and Isaac L. Chuang. 2017. Optimal Hamiltonian Simulation by Quantum Signal Processing. Physical Review Letters 118, 1 (2017), 010501. arXiv: 1606.02685Google ScholarCross Ref
- Guang Hao Low, Theodore J. Yoder, and Isaac L. Chuang. 2016. Methodology of Resonant Equiangular Composite Quantum Gates. Physical Review X 6, 4 (2016), 041067.Google Scholar
- arXiv: 1603.03996Google Scholar
- Frédéric Magniez, Ashwin Nayak, Jérémie Roland, and Miklos Santha. 2011.Google Scholar
- Search via Quantum Walk. SIAM Journal on Computing 40, 1 (2011), 142–164. arXiv: quant-ph/0608026 Google ScholarDigital Library
- Chris Marriott and John Watrous. 2005. Quantum Arthur–Merlin games. Computational Complexity 14, 2 (2005), 122–152. arXiv: cs/0506068 Google ScholarDigital Library
- Daniel Nagaj, Pawel Wocjan, and Yong Zhang. 2009. Fast Amplification of QMA. Quantum Information and Computation 9, 11&12 (2009), 1053–1068. Google ScholarDigital Library
- arXiv: 0904.1549Google Scholar
- David Poulin and Pawel Wocjan. 2009. Sampling from the Thermal Quantum Gibbs State and Evaluating Partition Functions with a Quantum Computer. Physical Review Letters 103, 22 (2009), 220502.Google ScholarCross Ref
- arXiv: 0905.2199Google Scholar
- Sushant Sachdeva and Nisheeth K. Vishnoi. 2014. Faster Algorithms via Approximation Theory. Found. Trends Theor. Comput. Sci. 9, 2 (2014), 125–210. Google ScholarDigital Library
- Lana Sheridan, Dmitri Maslov, and Michele Mosca. 2009. Approximating fractional time quantum evolution. Journal of Physics A: Mathematical and Theoretical 42, 18 (2009), 185302.Google ScholarCross Ref
- arXiv: 0810.3843Google Scholar
- Peter W. Shor. 1997. Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer. SIAM Journal on Computing 26, 5 (1997), 1484–1509. Google ScholarDigital Library
- Earlier version in FOCS’94. arXiv: quant-ph/9508027Google Scholar
- Peter W. Shor. 2003.Google Scholar
- Why Haven’t More Quantum Algorithms Been Found? Journal of the ACM 50, 1 (2003), 87–90. Google ScholarDigital Library
- Márió Szegedy. 2004.Google Scholar
- Quantum speed-up of Markov chain based algorithms. In Proceedings of the 45th IEEE Symposium on Foundations of Computer Science (FOCS). 32–41. arXiv: quant-ph/0401053 Google ScholarDigital Library
- Theodore J. Yoder, Guang Hao Low, and Isaac L. Chuang. 2014.Google Scholar
- Fixed-Point Quantum Search with an Optimal Number of Queries. Physical Review Letters 113, 21 (2014), 210501.Google Scholar
Index Terms
- Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics
Recommendations
Dequantizing the Quantum singular value transformation: hardness and applications to Quantum chemistry and the Quantum PCP conjecture
STOC 2022: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of ComputingThe Quantum Singular Value Transformation (QSVT) is a recent technique that gives a unified framework to describe most quantum algorithms discovered so far, and may lead to the development of novel quantum algorithms. In this paper we investigate the ...
Quantum correlation swapping
Quantum correlations (QCs), including quantum entanglement and those different, are important quantum resources and have attracted much attention recently. Quantum entanglement swapping as a kernel technique has already been applied to quantum repeaters ...
Quantum information processing beyond polarization encoding
GLSVLSI '11: Proceedings of the 21st edition of the great lakes symposium on Great lakes symposium on VLSII will present recent research results in optical one-way quantum computation and quantum information processing that I have been involved in. The first topic is an optimization of how to use the limited resources available in typical quantum-optics ...
Comments