Quantum process tomography with unsupervised learning and tensor networks

The impressive pace of advance of quantum technology calls for robust and scalable techniques for the characterization and validation of quantum hardware. Quantum process tomography, the reconstruction of an unknown quantum channel from measurement data, remains the quintessential primitive to completely characterize quantum devices. However, due to the exponential scaling of the required data and classical post-processing, its range of applicability is typically restricted to one- and two-qubit gates. Here, we present a new technique for performing quantum process tomography that addresses these issues by combining a tensor network representation of the channel with a data-driven optimization inspired by unsupervised machine learning. We demonstrate our technique through synthetically generated data for ideal one- and two-dimensional random quantum circuits of up to 10 qubits, and a noisy 5-qubit circuit, reaching process fidelities above 0.99 using only a limited set of single-qubit measurement samples and input states. Our results go far beyond state-of-the-art, providing a practical and timely tool for benchmarking quantum circuits in current and near-term quantum computers.

[1]  Andrew J. Ferris,et al.  Perfect Sampling with Unitary Tensor Networks , 2012, 1201.3974.

[2]  Travis S. Humble,et al.  Quantum supremacy using a programmable superconducting processor , 2019, Nature.

[3]  Kristan Temme,et al.  Supervised learning with quantum-enhanced feature spaces , 2018, Nature.

[4]  M. Mariantoni,et al.  Surface codes: Towards practical large-scale quantum computation , 2012, 1208.0928.

[5]  T. Ralph,et al.  Quantum process tomography of a controlled-NOT gate. , 2004, Physical review letters.

[6]  Roger G. Melko,et al.  Reconstructing quantum states with generative models , 2018, Nature Machine Intelligence.

[7]  Man-Duen Choi.  Completely positive linear maps on complex matrices , 1975 .

[8]  J Eisert,et al.  Fidelity Witnesses for Fermionic Quantum Simulations. , 2017, Physical review letters.

[9]  J. Preskill,et al.  Topological quantum memory , 2001, quant-ph/0110143.

[10]  A. Jamiołkowski.  Linear transformations which preserve trace and positive semidefiniteness of operators , 1972 .

[11]  Roger G. Melko,et al.  Machine-Learning Quantum States in the NISQ Era , 2019, Annual Review of Condensed Matter Physics.

[12]  M. Lukin,et al.  Probing many-body dynamics on a 51-atom quantum simulator , 2017, Nature.

[13]  Jens Eisert,et al.  Recovering quantum gates from few average gate fidelities , 2018, Physical review letters.

[14]  B. Lanyon,et al.  Observation of entangled states of a fully-controlled 20 qubit system , 2017, 1711.11092.

[15]  Stephen Becker,et al.  Quantum state tomography via compressed sensing. , 2009, Physical review letters.

[16]  L. C. G. Govia,et al.  Bootstrapping quantum process tomography via a perturbative ansatz , 2019, Nature Communications.

[17]  John Preskill,et al.  Quantum Computing in the NISQ era and beyond , 2018, Quantum.

[18]  Lei Wang,et al.  Scalable quantum tomography with fidelity estimation , 2017, Physical Review A.

[19]  Roger G. Melko,et al.  Integrating Neural Networks with a Quantum Simulator for State Reconstruction. , 2019, Physical review letters.

[20]  Roger G. Melko,et al.  Wave-function positivization via automatic differentiation , 2019, 1906.04654.

[21]  Matthias Troyer,et al.  Neural-network quantum state tomography , 2018 .

[22]  Lu-Ming Duan,et al.  Efficient representation of quantum many-body states with deep neural networks , 2017, Nature Communications.

[23]  R. Kosut,et al.  Efficient measurement of quantum dynamics via compressive sensing. , 2009, Physical review letters.

[24]  J. Eisert,et al.  Reliable quantum certification of photonic state preparations , 2014, Nature Communications.

[25]  Isaac L. Chuang,et al.  Prescription for experimental determination of the dynamics of a quantum black box , 1997 .

[26]  Yusuke Nomura,et al.  Constructing exact representations of quantum many-body systems with deep neural networks , 2018, Nature Communications.

[27]  Jun Wang,et al.  Unsupervised Generative Modeling Using Matrix Product States , 2017, Physical Review X.

[28]  Aaron C. E. Lee,et al.  Many-body localization in a quantum simulator with programmable random disorder , 2015, Nature Physics.

[29]  C. K. Andersen,et al.  Benchmarking Coherent Errors in Controlled-Phase Gates due to Spectator Qubits , 2020, 2005.05914.

[30]  Seth Lloyd,et al.  Quantum process tomography of the quantum Fourier transform. , 2004, The Journal of chemical physics.

[31]  Jens Eisert,et al.  Expressive power of tensor-network factorizations for probabilistic modeling, with applications from hidden Markov models to quantum machine learning , 2019, NeurIPS.

[32]  Lloyd C. L. Hollenberg,et al.  Experimental non-Markovian process characterisation and control on a quantum processor , 2020, 2004.14018.

[33]  M B Plenio,et al.  Scalable reconstruction of density matrices. , 2012, Physical review letters.

[34]  D. Gross,et al.  Efficient quantum state tomography. , 2010, Nature communications.

[35]  J. Gambetta,et al.  Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets , 2017, Nature.

[36]  Yi-Kai Liu,et al.  Direct fidelity estimation from few Pauli measurements. , 2011, Physical review letters.

[37]  Matthias Troyer,et al.  Solving the quantum many-body problem with artificial neural networks , 2016, Science.

[38]  J. Gambetta,et al.  Universal quantum gate set approaching fault-tolerant thresholds with superconducting qubits. , 2012, Physical review letters.

[39]  Jay M. Gambetta,et al.  Characterizing Quantum Gates via Randomized Benchmarking , 2011, 1109.6887.

[40]  Huaiyu Zhu.  On Information and Sufficiency , 1997 .

[41]  G M D'Ariano,et al.  Quantum tomography for measuring experimentally the matrix elements of an arbitrary quantum operation. , 2001, Physical review letters.

[42]  Dario Poletti,et al.  Tensor network based machine learning of non-Markovian quantum processes , 2020 .

[43]  C. F. Roos,et al.  Efficient tomography of a quantum many-body system , 2016, Nature Physics.

[44]  Ny,et al.  Quantum process tomography of a universal entangling gate implemented with Josephson phase qubits , 2009, 0910.1118.

[45]  P. Zoller,et al.  Self-verifying variational quantum simulation of lattice models , 2018, Nature.

[46]  I. McCulloch,et al.  Generic construction of efficient matrix product operators , 2016, 1611.02498.

[47]  J. Cirac,et al.  Neural-Network Quantum States, String-Bond States, and Chiral Topological States , 2017, 1710.04045.

[48]  J. Eisert,et al.  Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas , 2011, Nature Physics.

[49]  Lei Wang,et al.  Differentiable Programming Tensor Networks , 2019, Physical Review X.

[50]  Rainer Blatt,et al.  Characterizing large-scale quantum computers via cycle benchmarking , 2019, Nature Communications.

[51]  T. Monz,et al.  Process tomography of ion trap quantum gates. , 2006, Physical review letters.

[52]  A. H. Werner,et al.  Positive Tensor Network Approach for Simulating Open Quantum Many-Body Systems. , 2014, Physical review letters.

[53]  Andrew W. Cross,et al.  Correlated randomized benchmarking , 2018, 2003.02354.

[54]  David G. Cory,et al.  Tensor networks and graphical calculus for open quantum systems , 2011, Quantum Inf. Comput..

[55]  David Poulin,et al.  Practical characterization of quantum devices without tomography. , 2011, Physical review letters.

[56]  Steven T. Flammia,et al.  Efficient learning of quantum noise , 2019, Nature Physics.

[57]  Masoud Mohseni,et al.  Quantum approximate optimization of non-planar graph problems on a planar superconducting processor , 2020, 2004.04197.