可调耦合超导量子比特CZ门的校准与优化

量子计算在模拟量子系统和大数分解等特定问题上展现出比经典计算更强大的能力。然而，量子计算中作为信息载体的量子比特极易受到多种类型的错误影响，例如常见的比特翻转错误和相位翻转错误等。为了保护储存在量子比特中的信息，量子纠错技术引入额外的量子比特和特定的编码方案来检测和修正这些错误。量子纠错技术是实现大规模、容错量子计算机的关键，对于量子计算的实际应用和未来发展至关重要。

表面码作为量子纠错技术最具潜力的候选方案之一，其纠错性能随码距的扩大而指数级增加。通过一系列两比特门，表面码将一个逻辑量子比特上的信息分散存储在多个辅助量子比特上，使得单个量子比特的错误不会直接导致逻辑比特的错误。在超导量子计算平台上通常使用CZ门与特定的单比特门构造表面码所需要的两比特门。
CZ门是一种根据控制比特状态来决定目标比特上是否积累$\pi$相位的量子门。
本文介绍了可调耦合架构中利用$|11\rangle$-$|20\rangle$共振实现的CZ门方案。我们利用该方案中相位和泄漏分别只对耦合器和比特上的调制信号敏感的特点，极大简化了优化流程，并使用随机基准测试表征得到保真度为99.21\%的CZ门。

本文还提出了一种利用可调耦合器实现的泄漏抑制单元，该单元对芯片上的关联噪声具有显著的抑制效果。关联噪声是实现量子纠错道路上的根本性障碍，因为量子纠错的基本假设是错误之间不存在关联性。这种关联噪声来源于能量向非计算能级的泄漏，严重影响纠错性能。
该方案通过在可调耦合器上施加参数驱动信号，显著抑制了比特和耦合器上的关联错误。利用已标定的CZ门构造了个三比特的Z稳定子纠错线路，并将上述泄漏抑制单元加入到线路中，我们证明了该方案与量子纠错的兼容性。

[1] SHOR P W. Algorithms for quantum computation: discrete logarithms and factoring[C]//Proceedings 35th annual symposium on foundations of computer science. IEEE, 1994: 124-134.
[2] SHOR P W. Polynomial-time algorithms for prime factorization and discrete logarithms on aquantum computer[J]. SIAM Review, 1999, 41(2): 303-332.
[3] GROVER L K. Quantum mechanics helps in searching for a needle in a haystack[J]. PhysicalReview Letters, 1997, 79(2): 325.
[4] GROVER L K. A fast quantum mechanical algorithm for database search[C]//Proceedings ofthe twenty-eighth annual ACM symposium on Theory of computing. 1996: 212-219.
[5] WANG H Y, ZHANG W Y, YAO Z, et al. Interrelated thermalization and quantum criticalityin a lattice gauge simulator[J]. Physical Review Letters, 2023, 131(5): 050401.
[6] GUO Q, CHENG C, SUN Z H, et al. Observation of energy-resolved many-body localization[J]. Nature Physics, 2021, 17(2): 234-239.
[7] XU K, SUN Z H, LIU W, et al. Probing dynamical phase transitions with a superconductingquantum simulator[J]. Science Advances, 2020, 6(25): eaba4935.
[8] BLOCH F. Nuclear induction[J]. Physical Review, 1946, 70(7-8): 460.
[9] KRANTZ P, KJAERGAARD M, YAN F, et al. A Quantum Engineer’s Guide to SuperconductingQubits[J]. Applied Physics Reviews, 2019, 6(2): 021318.
[10] KITAEV A Y. Quantum computations: algorithms and error correction[J]. Russian MathematicalSurveys, 1997, 52(6): 1191.
[11] DAWSON C, NIELSEN M. The Solovay-Kitaev algorithm[J]. Quantum Information and Computation,2006, 6(1): 81-95.
[12] KELLY J, BARENDS R, FOWLER A G, et al. State preservation by repetitive error detectionin a superconducting quantum circuit[J]. Nature, 2015, 519(7541): 66-69.
[13] KITAEV A Y. Fault-tolerant quantum computation by anyons[J]. Annals of Physics, 2003, 303(1): 2-30.
[14] SHOR P W. Fault-tolerant quantum computation[C]//Proceedings of 37th conference on foundationsof computer science. IEEE, 1996: 56-65.
[15] BRAVYI S B, KITAEV A Y. Quantum codes on a lattice with boundary[A]. 1998. arXivpreprint quant-ph/9811052.
[16] FOWLER A G, MARIANTONI M, MARTINIS J M, et al. Surface codes: Towards practicallarge-scale quantum computation[J]. Physical Review A, 2012, 86(3): 032324.
[17] KITAEV A. Anyons in an exactly solved model and beyond[J]. Annals of Physics, 2006, 321(1): 2-111.
[18] RAUSSENDORF R, HARRINGTON J. Fault-tolerant quantum computation with high threshold in two dimensions[J]. Physical Review Letters, 2007, 98(19): 190504.
[19] RAUSSENDORF R, HARRINGTON J, GOYAL K. Topological fault-tolerance in cluster statequantum computation[J]. New Journal of Physics, 2007, 9(6): 199.
[20] SANK D, CHEN Z, KHEZRI M, et al. Measurement-induced state transitions in a superconductingqubit: Beyond the rotating wave approximation[J]. Physical Review Letters, 2016, 117(19): 190503.
[21] KHEZRI M, OPREMCAK A, CHEN Z, et al. Measurement-induced state transitions in a superconductingqubit: Within the rotating-wave approximation[J]. Physical Review Applied, 2023,20(5): 054008.
[22] TINKHAM M. Introduction to superconductivity[M]. Courier Corporation, 2004.
[23] GAO Y Y, LESTER B J, CHOU K S, et al. Entanglement of bosonic modes through an engineeredexchange interaction[J]. Nature, 2019, 566(7745): 509-512.
[24] ROSENBLUM S, GAO Y Y, REINHOLD P, et al. A CNOT gate between multiphoton qubitsencoded in two cavities[J]. Nature communications, 2018, 9(1): 652.
[25] RIGETTI C, DEVORET M. Fully microwave-tunable universal gates in superconducting qubitswith linear couplings and fixed transition frequencies[J]. Physical Review B, 2010, 81(13):134507.
[26] CHOW J M, CÓRCOLES A D, GAMBETTA J M, et al. Simple all-microwave entangling gatefor fixed-frequency superconducting qubits[J]. Physical Review Letters, 2011, 107(8): 080502.
[27] MAGESAN E, GAMBETTA J M. Effective Hamiltonian models of the cross-resonance gate[J]. Physical Review A, 2020, 101(5): 052308.
[28] TRIPATHI V, KHEZRI M, KOROTKOV A N. Operation and intrinsic error budget of a twoqubitcross-resonance gate[J]. Physical Review A, 2019, 100(1): 012301.
[29] CALDWELL S A, DIDIER N, RYAN C A, et al. Parametrically Activated Entangling GatesUsing Transmon Qubits[J]. Physical Review Applied, 2018, 10(3): 034050.
[30] SETE E A, DIDIER N, CHEN A Q, et al. Parametric-resonance entangling gates with a tunablecoupler[J]. Physical Review Applied, 2021, 16(2): 024050.
[31] CHU J, LI D, YANG X, et al. Realization of superadiabatic two-qubit gates using parametricmodulation in superconducting circuits[J]. Physical Review Applied, 2020, 13(6): 064012.
[32] REAGOR M, OSBORN C B, TEZAK N, et al. Demonstration of universal parametric entanglinggates on a multi-qubit lattice[J]. Science Advances, 2018, 4(2): eaao3603.
[33] MARTINIS J M, GELLER M R. Fast adiabatic qubit gates using only 𝜎 z control[J]. PhysicalReview A, 2014, 90(2): 022307.
[34] SUNG Y, DING L, BRAUMÜLLER J, et al. Realization of high-fidelity cz and z z-free iswapgates with a tunable coupler[J]. Physical Review X, 2021, 11(2): 021058.
[35] REED M D, JOHNSON B R, HOUCK A A, et al. Fast reset and suppressing spontaneousemission of a superconducting qubit[J]. Applied Physics Letters, 2010, 96(20).
[36] SETE E A, MARTINIS J M, KOROTKOV A N. Quantum theory of a bandpass Purcell filterfor qubit readout[J]. Physical Review A, 2015, 92(1): 012325.
[37] BLAIS A, HUANG R S, WALLRAFF A, et al. Cavity quantum electrodynamics for superconductingelectrical circuits: An architecture for quantum computation[J]. Physical Review A,2004, 69(6): 062320.
[38] JEFFREY E, SANK D, MUTUS J, et al. Fast accurate state measurement with superconductingqubits[J]. Physical Review Letters, 2014, 112(19): 190504.
[39] CHEN Z. Metrology of quantum control and measurement in superconducting qubits[M]. Universityof California, Santa Barbara, 2018.
[40] MOTZOI F, GAMBETTA J M, REBENTROST P, et al. Simple pulses for elimination of leakagein weakly nonlinear qubits[J]. Physical Review Letters, 2009, 103(11): 110501.
[41] LUCERO E, KELLY J, BIALCZAK R C, et al. Reduced phase error through optimized controlof a superconducting qubit[J]. Physical Review A, 2010, 82(4): 042339.
[42] REED M. Entanglement and quantum error correction with superconducting qubits[M]. Lulu.com, 2013.
[43] EMERSON J, ALICKI R, ŻYCZKOWSKI K. Scalable noise estimation with random unitaryoperators[J]. Journal of Optics B: Quantum and Semiclassical Optics, 2005, 7(10): S347.
[44] KNILL E, LEIBFRIED D, REICHLE R, et al. Randomized benchmarking of quantum gates[J]. Physical Review A, 2008, 77(1): 012307.
[45] MAGESAN E, GAMBETTA J M, EMERSON J. Scalable and robust randomized benchmarkingof quantum processes[J]. Physical Review Letters, 2011, 106(18): 180504.
[46] DANKERT C, CLEVE R, EMERSON J, et al. Exact and approximate unitary 2-designs andtheir application to fidelity estimation[J]. Physical Review A, 2009, 80(1): 012304.
[47] BARENDS R, KELLY J, MEGRANT A, et al. Superconducting quantum circuits at the surfacecode threshold for fault tolerance[J]. Nature, 2014, 508(7497): 500-503.
[48] TERHAL B M. Quantum error correction for quantum memories[J]. Reviews of ModernPhysics, 2015, 87(2): 307.
[49] PRESKILL J. Quantum computing in the NISQ era and beyond[J]. Quantum, 2018, 2: 79.
[50] DIVINCENZO D P, IBM. The Physical Implementation of Quantum Computation[J].Fortschritte der Physik, 2000, 48(9-11): 771-783.
[51] FOWLER A G, MARIANTONI M, MARTINIS J M, et al. Surface Codes: Towards PracticalLarge-Scale Quantum Computation[J]. Physical Review A, 2012, 86(3): 032324.
[52] CHEN Z, SATZINGER K J, ATALAYA J, et al. Exponential Suppression of Bit or Phase Errorswith Cyclic Error Correction[J]. Nature, 2021, 595(7867): 383-387.
[53] KRINNER S, LACROIX N, REMM A, et al. Realizing repeated quantum error correction in adistance-three surface code[J]. Nature, 2022, 605(7911): 669-674.
[54] ZHAO Y, YE Y, HUANG H L, et al. Realization of an error-correcting surface code withsuperconducting qubits[J]. Physical Review Letters, 2022, 129(3): 030501.
[55] ACHARYA R, ALEINER I, ALLEN R, et al. Suppressing Quantum Errors by Scaling a SurfaceCode Logical Qubit[J]. Nature, 2023, 614(7949): 676-681.
[56] FOWLER A G. Coping with Qubit Leakage in Topological Codes[J]. Physical Review A, 2013,88(4): 042308.
[57] GHOSH J, FOWLER A G, MARTINIS J M, et al. Understanding the effects of leakage insuperconducting quantum-error-detection circuits[J]. Physical Review A, 2013, 88(6): 062329.
[58] MCEWEN M, KAFRI D, CHEN Z, et al. Removing Leakage-Induced Correlated Errors inSuperconducting Quantum Error Correction[J]. Nature Communications, 2021, 12(1): 1761.
[59] MIAO K C, MCEWEN M, ATALAYA J, et al. Overcoming Leakage in Quantum Error Correction[J]. Nature Physics, 2023: 1-7.
[60] BARENDS R, KELLY J, MEGRANT A, et al. Superconducting Quantum Circuits at the SurfaceCode Threshold for Fault Tolerance[J]. Nature, 2014, 508(7497): 500-503.
[61] BLUVSTEIN D, EVERED S J, GEIM A A, et al. Logical quantum processor based on reconfigurableatom arrays[J]. Nature, 2024, 626(7997): 58-65.
[62] EVERED S J, BLUVSTEIN D, KALINOWSKI M, et al. High-fidelity parallel entangling gateson a neutral-atom quantum computer[J]. Nature, 2023, 622(7982): 268-272.
[63] XUE X, RUSS M, SAMKHARADZE N, et al. Quantum logic with spin qubits crossing thesurface code threshold[J]. Nature, 2022, 601(7893): 343-347.
[64] GHOSH J, FOWLER A G. Leakage-Resilient Approach to Fault-Tolerant Quantum Computingwith Superconducting Elements[J]. Physical Review A, 2015, 91(2): 020302.
[65] BATTISTEL F, VARBANOV B, TERHAL B. Hardware-Efficient Leakage-Reduction Schemefor Quantum Error Correction with Superconducting Transmon Qubits[J]. PRX Quantum, 2021,2(3): 030314.
[66] ZHOU Y, ZHANG Z, YIN Z, et al. Rapid and Unconditional Parametric Reset Protocol forTunable Superconducting Qubits[J]. Nature Communications, 2021, 12(1): 5924.
[67] MARQUES J F, ALI H, VARBANOV B M, et al. All-Microwave Leakage Reduction Units forQuantum Error Correction with Superconducting Transmon Qubits[J]. Physical Review Letters,2023, 130(25): 250602.
[68] LACROIX N, HOFELE L, REMM A, et al. Fast Flux-Activated Leakage Reduction for SuperconductingQuantum Circuits[A]. 2023. arXiv:2309.07060.
[69] YANG X, CHU J, GUO Z, et al. Coupler-Assisted Leakage Reduction for Scalable QuantumError Correction with Superconducting Qubits[A]. 2024. arXiv:2403.16155.
[70] XU Y, CHU J, YUAN J, et al. High-fidelity, high-scalability two-qubit gate scheme for superconductingqubits[J]. Physical Review Letters, 2020, 125(24): 240503.
[71] CHU J, YAN F. Coupler-assisted controlled-phase gate with enhanced adiabaticity[J]. PhysicalReview Applied, 2021, 16(5): 054020.