Qubit-efficient entanglement spectroscopy using qubit resets

1Department of Computer Science, The University of Texas at Austin, Austin, TX 78712, USA
2Computer, Computational, and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Abstract

One strategy to fit larger problems on NISQ devices is to exploit a tradeoff between circuit width and circuit depth. Unfortunately, this tradeoff still limits the size of tractable problems since the increased depth is often not realizable before noise dominates. Here, we develop $\textit{qubit-efficient}$ quantum algorithms for entanglement spectroscopy which avoid this tradeoff. In particular, we develop algorithms for computing the trace of the $n$-th power of the density operator of a quantum system, $Tr(\rho^n)$, (related to the Rényi entropy of order $n$) that use fewer qubits than any previous efficient algorithm while achieving similar performance in the presence of noise, thus enabling spectroscopy of larger quantum systems on NISQ devices. Our algorithms, which require a number of qubits independent of $n$, are variants of previous algorithms with width proportional to $n$, an asymptotic difference. The crucial ingredient in these new algorithms is the ability to measure and reinitialize subsets of qubits in the course of the computation, allowing us to reuse qubits and increase the circuit depth without suffering the usual noisy consequences. We also introduce the notion of $\textit{effective circuit depth}$ as a generalization of standard circuit depth suitable for circuits with qubit resets. This tool helps explain the noise-resilience of our qubit-efficient algorithms and should aid in designing future algorithms. We perform numerical simulations to compare our algorithms to the original variants and show they perform similarly when subjected to noise. Additionally, we experimentally implement one of our qubit-efficient algorithms on the Honeywell System Model H0, estimating $Tr(\rho^n)$ for larger n than possible with previous algorithms.

In the near- to medium-term, during the so-called Noisy Intermediate-Scale Quantum (NISQ) era, quantum devices don’t have many qubits and can’t stay coherent for long. As a result, only quantum circuits of limited width (the number of qubits) and depth (similar to the runtime) can be successfully executed. For many applications, larger problem size corresponds to larger circuits. Therefore, quantum computers will remain limited to small problems for a significant time unless new solutions are found.

One strategy to fit larger problems on NISQ devices is to exploit a tradeoff between circuit width and circuit depth. Unfortunately, since both of these resources are limited, this tradeoff still limits size of tractable problems.

Here, we develop qubit-efficient quantum algorithms for the task of Entanglement Spectroscopy which avoid this tradeoff.

In this application, the goal is to learn the structure of the entanglement between subsystems of a larger quantum system. Entanglement spectroscopy will be important in order to process the output of one of the most promising applications of quantum computers, quantum simulation of many-body systems.

In particular, we develop algorithms for Entanglement Spectroscopy that use asymptotically fewer qubits than any previous (efficient) algorithms while achieving similar performance in the presence of noise. Thus, our algorithms avoid the usual tradeoff and enable spectroscopy of larger quantum systems on NISQ devices than previously possible.

The crucial ingredient in these new algorithms is the ability to measure and reinitialize subsets of qubits in the course of the computation. By carefully arranging these qubits resets, we are able to reuse qubits and increase the circuit depth without suffering the usual noisy consequences.

We perform experiments to test our algorithms on both simulated and real quantum hardware. We experimentally implement one of our qubit-efficient algorithms on the 6-qubit Honeywell System Model H0. In these experiments, we perform spectroscopy for larger parameters than possible with any previous algorithm on this size device. We also run numerical simulations using IBM’s Qiskit to test the noise-resilience of our algorithms and compare them to previous algorithms.

Finally, we introduce the notion of effective circuit depth as a generalization of standard circuit depth that is suitable for circuits with qubit resets. This tool helps explain the noise-resilience of our qubit-efficient algorithms, while standard circuit depth does not, and should aid in designing future algorithms.

► References

[1] N. Abdessaied, R. Wille, M. Soeken, and R. Drechsler. Reducing the depth of quantum circuits using additional circuit lines. In G.W. Dueck and D.M. Miller, editors, Reversible Computation. RC 2013., page 221–233, Berlin, Heidelberg, 2013. Springer-Verlag. 10.1007/​978-3-642-38986-3_18.
https:/​/​doi.org/​10.1007/​978-3-642-38986-3_18

[2] Héctor Abraham et al. Qiskit: An open-source framework for quantum computing, 2019. URL https:/​/​github.com/​Qiskit.
https:/​/​github.com/​Qiskit

[3] Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral. Entanglement in many-body systems. Rev. Mod. Phys., 80 (2): 517, 2008. 10.1103/​RevModPhys.80.517.
https:/​/​doi.org/​10.1103/​RevModPhys.80.517

[4] Galit Anikeeva, Isaac H Kim, and Patrick Hayden. Recycling qubits in near-term quantum computers. Physical Review A, 103 (4): 042613, 2021. 10.1103/​PhysRevA.103.042613.
https:/​/​doi.org/​10.1103/​PhysRevA.103.042613

[5] Javier Argüello-Luengo, Alejandro González-Tudela, Tao Shi, Peter Zoller, and J Ignacio Cirac. Analogue quantum chemistry simulation. Nature, 574 (7777): 215–218, Oct 2019. 10.1038/​s41586-019-1614-4.
https:/​/​doi.org/​10.1038/​s41586-019-1614-4

[6] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando G.S.L. Brandao, David A. Buell, et al. Quantum supremacy using a programmable superconducting processor. Nature, 574 (7779): 505–510, 2019. 10.1038/​s41586-019-1666-5.
https:/​/​doi.org/​10.1038/​s41586-019-1666-5

[7] Costin Bădescu, Ryan O'Donnell, and John Wright. Quantum state certification. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 503–514, 2019. 10.1145/​3313276.3316344.
https:/​/​doi.org/​10.1145/​3313276.3316344

[8] X. Bonet-Monroig, R. Sagastizabal, M. Singh, and T.E. O'Brien. Low-cost error mitigation by symmetry verification. Physical Review A, 98 (6): 062339, 2018. 10.1103/​PhysRevA.98.062339.
https:/​/​doi.org/​10.1103/​PhysRevA.98.062339

[9] Anne Broadbent and Elham Kashefi. Parallelizing quantum circuits. Theoretical Computer Science, 410 (26): 2489–2510, 2009. 10.1016/​j.tcs.2008.12.046.
https:/​/​doi.org/​10.1016/​j.tcs.2008.12.046

[10] Todd A Brun. Measuring polynomial functions of states. arXiv:quant-ph/​0401067v2, 2004.
arXiv:quant-ph/0401067v2

[11] Harry Buhrman, Richard Cleve, John Watrous, and Ronald de Wolf. Quantum fingerprinting. Phys. Rev. Lett., 87: 167902, Sep 2001. 10.1103/​PhysRevLett.87.167902.
https:/​/​doi.org/​10.1103/​PhysRevLett.87.167902

[12] Pasquale Calabrese and Alexandre Lefevre. Entanglement spectrum in one-dimensional systems. Phys. Rev. A, 78: 032329, Sep 2008. 10.1103/​PhysRevA.78.032329.
https:/​/​doi.org/​10.1103/​PhysRevA.78.032329

[13] Lukasz Cincio, Yiğit Subaşı, Andrew T Sornborger, and Patrick J Coles. Learning the quantum algorithm for state overlap. New Journal of Physics, 20 (11): 113022, 2018. 10.1088/​1367-2630/​aae94a.
https:/​/​doi.org/​10.1088/​1367-2630/​aae94a

[14] Antonio D Córcoles, Abhinav Kandala, Ali Javadi-Abhari, Douglas T McClure, Andrew W Cross, Kristan Temme, Paul D Nation, Matthias Steffen, and JM Gambetta. Challenges and opportunities of near-term quantum computing systems. Proceedings of the IEEE, 2019. 10.1109/​JPROC.2019.2954005.
https:/​/​doi.org/​10.1109/​JPROC.2019.2954005

[15] M. Dalmonte, B. Vermersch, and P. Zoller. Quantum simulation and spectroscopy of entanglement Hamiltonians. Nature Physics, 14 (8): 827–831, May 2018. 10.1038/​s41567-018-0151-7.
https:/​/​doi.org/​10.1038/​s41567-018-0151-7

[16] G. De Chiara, L. Lepori, M. Lewenstein, and A. Sanpera. Entanglement spectrum, critical exponents, and order parameters in quantum spin chains. Phys. Rev. Lett., 109: 237208, Dec 2012. 10.1103/​PhysRevLett.109.237208.
https:/​/​doi.org/​10.1103/​PhysRevLett.109.237208

[17] D.J. Egger, M. Werninghaus, M. Ganzhorn, G. Salis, A. Fuhrer, P. Mueller, and S. Filipp. Pulsed reset protocol for fixed-frequency superconducting qubits. Phys. Rev. Applied, 10: 044030, Oct 2018. 10.1103/​PhysRevApplied.10.044030.
https:/​/​doi.org/​10.1103/​PhysRevApplied.10.044030

[18] Suguru Endo, Zhenyu Cai, Simon C Benjamin, and Xiao Yuan. Hybrid quantum-classical algorithms and quantum error mitigation. Journal of the Physical Society of Japan, 90 (3): 032001, 2021. 10.7566/​JPSJ.90.032001.
https:/​/​doi.org/​10.7566/​JPSJ.90.032001

[19] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm. arXiv:1411.4028v1 [quant-ph], 2014.
arXiv:1411.4028v1

[20] Lukasz Fidkowski. Entanglement spectrum of topological insulators and superconductors. Phys. Rev. Lett., 104: 130502, Apr 2010. 10.1103/​PhysRevLett.104.130502.
https:/​/​doi.org/​10.1103/​PhysRevLett.104.130502

[21] Michael Foss-Feig, David Hayes, Joan M Dreiling, Caroline Figgatt, John P Gaebler, Steven A Moses, Juan M Pino, and Andrew C Potter. Holographic quantum algorithms for simulating correlated spin systems. Phys. Rev. Research, 3 (3): 033002, 2021. 10.1103/​PhysRevResearch.3.033002.
https:/​/​doi.org/​10.1103/​PhysRevResearch.3.033002

[22] Steph Foulds, Viv Kendon, and Tim Spiller. The controlled SWAP test for determining quantum entanglement. Quantum Science and Technology, 6 (3): 035002, 2021. 10.1088/​2058-9565/​abe458.
https:/​/​doi.org/​10.1088/​2058-9565/​abe458

[23] Juan Carlos Garcia-Escartin and Pedro Chamorro-Posada. SWAP test and Hong-Ou-Mandel effect are equivalent. Phys. Rev. A, 87: 052330, May 2013. 10.1103/​PhysRevA.87.052330.
https:/​/​doi.org/​10.1103/​PhysRevA.87.052330

[24] K. Geerlings, Z. Leghtas, I.M. Pop, S. Shankar, L. Frunzio, R.J. Schoelkopf, M. Mirrahimi, and M.H. Devoret. Demonstrating a driven reset protocol for a superconducting qubit. Phys. Rev. Lett., 110: 120501, Mar 2013. 10.1103/​PhysRevLett.110.120501.
https:/​/​doi.org/​10.1103/​PhysRevLett.110.120501

[25] Daniel Gottesman and Isaac Chuang. Quantum digital signatures. arXiv:quant-ph/​0105032v2, 2001.
arXiv:quant-ph/0105032v2

[26] Maria Hermanns. Entanglement in topological systems. arXiv:1702.01525v1 [cond-mat.str-el], 2017.
arXiv:1702.01525v1

[27] William Huggins, Piyush Patil, Bradley Mitchell, K Birgitta Whaley, and E Miles Stoudenmire. Towards quantum machine learning with tensor networks. Quantum Science and Technology, 4 (2): 024001, jan 2019. 10.1088/​2058-9565/​aaea94.
https:/​/​doi.org/​10.1088/​2058-9565/​aaea94

[28] Rajibul Islam, Ruichao Ma, Philipp M. Preiss, M. Eric Tai, Alexander Lukin, Matthew Rispoli, and Markus Greiner. Measuring entanglement entropy in a quantum many-body system. Nature, 528 (7580): 77–83, Dec 2015. 10.1038/​nature15750.
https:/​/​doi.org/​10.1038/​nature15750

[29] Sonika Johri, Damian S. Steiger, and Matthias Troyer. Entanglement spectroscopy on a quantum computer. Phys. Rev. B, 96: 195136, Nov 2017. 10.1103/​PhysRevB.96.195136.
https:/​/​doi.org/​10.1103/​PhysRevB.96.195136

[30] Christian Kokail, Rick van Bijnen, Andreas Elben, Benoı̂t Vermersch, and Peter Zoller. Entanglement hamiltonian tomography in quantum simulation. Nature Physics, pages 936–942, 2021. 10.1038/​s41567-021-01260-w.
https:/​/​doi.org/​10.1038/​s41567-021-01260-w

[31] Hui Li and F.D.M. Haldane. Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-Abelian fractional quantum Hall effect states. Phys. Rev. Lett., 101: 010504, Jul 2008. 10.1103/​PhysRevLett.101.010504.
https:/​/​doi.org/​10.1103/​PhysRevLett.101.010504

[32] Ying Li and Simon C Benjamin. Efficient variational quantum simulator incorporating active error minimization. Physical Review X, 7 (2): 021050, 2017. 10.1103/​PhysRevX.7.021050.
https:/​/​doi.org/​10.1103/​PhysRevX.7.021050

[33] N.M. Linke, S. Johri, C. Figgatt, K.A. Landsman, A.Y. Matsuura, and C. Monroe. Measuring the Rényi entropy of a two-site Fermi-Hubbard model on a trapped ion quantum computer. Phys. Rev. A, 98: 052334, Nov 2018. 10.1103/​PhysRevA.98.052334.
https:/​/​doi.org/​10.1103/​PhysRevA.98.052334

[34] Jin-Guo Liu, Yi-Hong Zhang, Yuan Wan, and Lei Wang. Variational quantum eigensolver with fewer qubits. Phys. Rev. Research, 1: 023025, Sep 2019. 10.1103/​PhysRevResearch.1.023025.
https:/​/​doi.org/​10.1103/​PhysRevResearch.1.023025

[35] Seth Lloyd, Masoud Mohseni, and Patrick Rebentrost. Quantum principal component analysis. Nature Physics, 10 (9): 631–633, 2014. 10.1038/​nphys3029.
https:/​/​doi.org/​10.1038/​nphys3029

[36] P. Magnard, P. Kurpiers, B. Royer, T. Walter, J.-C. Besse, S. Gasparinetti, M. Pechal, J. Heinsoo, S. Storz, A. Blais, and A. Wallraff. Fast and unconditional all-microwave reset of a superconducting qubit. Phys. Rev. Lett., 121: 060502, Aug 2018. 10.1103/​PhysRevLett.121.060502.
https:/​/​doi.org/​10.1103/​PhysRevLett.121.060502

[37] Jarrod R McClean, Jonathan Romero, Ryan Babbush, and Alán Aspuru-Guzik. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics, 18 (2): 023023, feb 2016. 10.1088/​1367-2630/​18/​2/​023023.
https:/​/​doi.org/​10.1088/​1367-2630/​18/​2/​023023

[38] D.T. McClure, Hanhee Paik, L.S. Bishop, M. Steffen, Jerry M. Chow, and Jay M. Gambetta. Rapid driven reset of a qubit readout resonator. Phys. Rev. Applied, 5: 011001, Jan 2016. 10.1103/​PhysRevApplied.5.011001.
https:/​/​doi.org/​10.1103/​PhysRevApplied.5.011001

[39] Tianyi Peng, Aram W Harrow, Maris Ozols, and Xiaodi Wu. Simulating large quantum circuits on a small quantum computer. Phys. Rev. Lett., 125 (15): 150504, 2020. 10.1103/​PhysRevLett.125.150504.
https:/​/​doi.org/​10.1103/​PhysRevLett.125.150504

[40] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J Love, Alán Aspuru-Guzik, and Jeremy L O’brien. A variational eigenvalue solver on a photonic quantum processor. Nature Communications, 5 (1): 4213, Jul 2014. 10.1038/​ncomms5213.
https:/​/​doi.org/​10.1038/​ncomms5213

[41] J.M. Pino, J.M. Dreiling, C. Figgatt, J.P. Gaebler, S.A. Moses, M.S. Allman, C.H. Baldwin, M. Foss-Feig, D. Hayes, K. Mayer, C. Ryan-Anderson, and B. Neyenhuis. Demonstration of the trapped-ion quantum CCD computer architecture. Nature, 592 (7853): 209–213, 2021. 10.1038/​s41586-021-03318-4.
https:/​/​doi.org/​10.1038/​s41586-021-03318-4

[42] John Preskill. Quantum Computing in the NISQ era and beyond. Quantum, 2: 79, August 2018. 10.22331/​q-2018-08-06-79.
https:/​/​doi.org/​10.22331/​q-2018-08-06-79

[43] Emil Prodan, Taylor L. Hughes, and B. Andrei Bernevig. Entanglement spectrum of a disordered topological Chern insulator. Phys. Rev. Lett., 105: 115501, Sep 2010. 10.1103/​PhysRevLett.105.115501.
https:/​/​doi.org/​10.1103/​PhysRevLett.105.115501

[44] Arthur G Rattew, Yue Sun, Pierre Minssen, and Marco Pistoia. Quantum simulation of Galton machines using mid-circuit measurement and reuse. arXiv:2009.06601v1 [quant-ph], 2020.
arXiv:2009.06601v1

[45] M.D. Reed, B.R. Johnson, A.A. Houck, L. DiCarlo, J.M. Chow, D.I. Schuster, L. Frunzio, and R.J. Schoelkopf. Fast reset and suppressing spontaneous emission of a superconducting qubit. Applied Physics Letters, 96 (20): 203110, 2010. 10.1063/​1.3435463.
https:/​/​doi.org/​10.1063/​1.3435463

[46] Diego Ristè, Luke CG Govia, Brian Donovan, Spencer D Fallek, William D Kalfus, Markus Brink, Nicholas T Bronn, and Thomas A Ohki. Real-time processing of stabilizer measurements in a bit-flip code. npj Quantum Information, 6 (1): 1–6, 2020. 10.1038/​s41534-020-00304-y.
https:/​/​doi.org/​10.1038/​s41534-020-00304-y

[47] Benjamin Schumacher and Michael D. Westmoreland. Locality and information transfer in quantum operations. Quantum Information Processing, 4 (1): 13–34, Feb 2005. 10.1007/​s11128-004-3193-y.
https:/​/​doi.org/​10.1007/​s11128-004-3193-y

[48] Benjamin Schumacher and Michael D. Westmoreland. Isolation and information flow in quantum dynamics. Foundations of Physics, 42 (7): 926–931, 2012. 10.1007/​s10701-012-9651-y.
https:/​/​doi.org/​10.1007/​s10701-012-9651-y

[49] H. Francis Song, Stephan Rachel, Christian Flindt, Israel Klich, Nicolas Laflorencie, and Karyn Le Hur. Bipartite fluctuations as a probe of many-body entanglement. Phys. Rev. B, 85: 035409, Jan 2012. 10.1103/​PhysRevB.85.035409.
https:/​/​doi.org/​10.1103/​PhysRevB.85.035409

[50] Yiğit Subaşı, Lukasz Cincio, and Patrick J Coles. Entanglement spectroscopy with a depth-two quantum circuit. Journal of Physics A: Mathematical and Theoretical, 52 (4): 044001, 2019. 10.1088/​1751-8121/​aaf54d.
https:/​/​doi.org/​10.1088/​1751-8121/​aaf54d

[51] Sathyawageeswar Subramanian and Min-Hsiu Hsieh. Quantum algorithm for estimating Renyi entropies of quantum states. arXiv:1908.05251v1 [quant-ph], 2019.
arXiv:1908.05251v1

[52] Brian Swingle and T. Senthil. Geometric proof of the equality between entanglement and edge spectra. Phys. Rev. B, 86: 045117, Jul 2012. 10.1103/​PhysRevB.86.045117.
https:/​/​doi.org/​10.1103/​PhysRevB.86.045117

[53] Kristan Temme, Sergey Bravyi, and Jay M Gambetta. Error mitigation for short-depth quantum circuits. Physical review letters, 119 (18): 180509, 2017. 10.1103/​PhysRevLett.119.180509.
https:/​/​doi.org/​10.1103/​PhysRevLett.119.180509

Cited by

[1] Youle Wang, Guangxi Li, and Xin Wang, "Variational Quantum Gibbs State Preparation with a Truncated Taylor Series", Physical Review Applied 16 5, 054035 (2021).

[2] Piotr Czarnik, Andrew Arrasmith, Lukasz Cincio, and Patrick J. Coles, "Qubit-efficient exponential suppression of errors", arXiv:2102.06056.

[3] Kun Wang, Yu-Ao Chen, and Xin Wang, "Measurement Error Mitigation via Truncated Neumann Series", arXiv:2103.13856.

[4] Michael Foss-Feig, Stephen Ragole, Andrew Potter, Joan Dreiling, Caroline Figgatt, John Gaebler, Alex Hall, Steven Moses, Juan Pino, Ben Spaun, Brian Neyenhuis, and David Hayes, "Entanglement from tensor networks on a trapped-ion QCCD quantum computer", arXiv:2104.11235.

[5] Maria Kieferova, Ortiz Marrero Carlos, and Nathan Wiebe, "Quantum Generative Training Using Rényi Divergences", arXiv:2106.09567.

[6] Kok Chuan Tan and Tyler Volkoff, "Variational quantum algorithms to estimate rank, quantum entropies, fidelity, and Fisher information via purity minimization", Physical Review Research 3 3, 033251 (2021).

[7] Kun Wang, Yu-Ao Chen, and Xin Wang, "Mitigating Quantum Errors via Truncated Neumann Series", arXiv:2111.00691.

The above citations are from Crossref's cited-by service (last updated successfully 2021-12-08 17:17:47) and SAO/NASA ADS (last updated successfully 2021-12-08 17:17:48). The list may be incomplete as not all publishers provide suitable and complete citation data.