ISSN 0253-2778

CN 34-1054/N

Open AccessOpen Access JUSTC Physics 06 September 2022

Pure state tomography with adaptive Pauli measurements

Cite this:
https://doi.org/10.52396/JUSTC-2022-0037
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  • Author Bio:

    Xiangrui Meng is currently a graduate student under the tutelage of Prof. Zhensheng Yuan at the University of Science and Technology of China. His research interests focus on quantum information and quantum computation

    Zhensheng Yuan is now a Professor of Physics at the Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China (USTC), and a winner of the National Science Fund for Outstanding Youth. He received a B.Sc. and a Ph.D. from USTC in 1998 and 2003, respectively. During 2006 to 2011, he had been working at the Heidelberg University as a PostDoc, an Alexander von Humboldt Fellow, and a senior scientist (when he was a CoPI of a couple of projects) successively. He took his professorship at USTC in 2011. His research field is quantum manipulation of light and cold atoms. Highlights of his research achievements include the quantum simulation of mechanisms in LGT and the manipulation of atomic spin entanglements in optical lattices. He has about 70 publications in peer-reviewed journals including Nature, Science, Nature Physics, and Physical Review Letters

  • Corresponding author: E-mail: yuanzs@ustc.edu.cn
  • Received Date: 19 February 2022
  • Accepted Date: 13 May 2022
  • Available Online: 06 September 2022
  • Quantum state tomography provides a key tool for validating and fully exploiting quantum resources. However, current protocols of pure-state informationally-complete (PS-IC) measurement settings generally involve various multi-qubit gates or complex quantum algorithms, which are not practical for large systems. In this study, we present an adaptive approach to $N$-qubit pure-state tomography with Pauli measurements. First, projective measurements on each qubit in the Z-direction were implemented to determine the amplitude of each base of the target state. Then, a set of Pauli measurement settings was recursively deduced by the Z-measurement results, which can be used to determine the phase of each base. The number of required measurement settings is $O(N)$ for certain quantum states, including cluster and W states. Finally, we numerically verified the feasibility of our strategy by reconstructing a 1-D chain state using a neural network algorithm.
    First, projective measurements on each qubit in the Z-direction were implemented to determine the amplitude of each base of the target state. Then, a set of Pauli measurement settings was recursively deduced by the Z-measurement results, which can be used to determine the phase of each base.
    Quantum state tomography provides a key tool for validating and fully exploiting quantum resources. However, current protocols of pure-state informationally-complete (PS-IC) measurement settings generally involve various multi-qubit gates or complex quantum algorithms, which are not practical for large systems. In this study, we present an adaptive approach to $N$-qubit pure-state tomography with Pauli measurements. First, projective measurements on each qubit in the Z-direction were implemented to determine the amplitude of each base of the target state. Then, a set of Pauli measurement settings was recursively deduced by the Z-measurement results, which can be used to determine the phase of each base. The number of required measurement settings is $O(N)$ for certain quantum states, including cluster and W states. Finally, we numerically verified the feasibility of our strategy by reconstructing a 1-D chain state using a neural network algorithm.
    • We present an adaptive strategy for the design of the PS-IC measurement settings using Pauli measurements.
    • The number of required measurement settings is O(N) for certain quantum states, including cluster and W states.
    • We numerically verified the feasibility of the strategy by reconstructing a 10-qubit 1-D chain state using the restricted Boltzmann machine (RBM) model.

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  • [1]
    Torlai G, Melko R G. Machine-learning quantum states in the NISQ era. Annual Review of Condensed Matter Physics, 2020, 11: 325–344. doi: 10.1146/annurev-conmatphys-031119-050651
    [2]
    Nielsen M A, Chuang I L. Quantum Computation and Quantum Information. Cambridge, UK: Cambridge University Press, 2000.
    [3]
    Prugovečki E. Information-theoretical aspects of quantum measurement. International Journal of Theoretical Physics, 1977, 16 (5): 321–331. doi: 10.1007/BF01807146
    [4]
    Busch P. Informationally complete sets of physical quantities. International Journal of Theoretical Physics, 1991, 30: 1217–1227. doi: 10.1007/BF00671008
    [5]
    Heinosaari T, Mazzarella L, Wolf M M. Quantum tomography under prior information. Communications in Mathematical Physics, 2013, 318 (2): 355–374. doi: 10.1007/s00220-013-1671-8
    [6]
    Chen J, Dawkins H, Ji Z, et al. Uniqueness of quantum states compatible with given measurement results. Physical Review A, 2013, 88: 012109. doi: 10.1103/PhysRevA.88.012109
    [7]
    Flammia S T, Silberfarb A, Caves C M. Minimal informationally complete measurements for pure states. Foundations of Physics, 2005, 35 (12): 1985–2006. doi: 10.1007/s10701-005-8658-z
    [8]
    Goyeneche D, Cañas G, Etcheverry S, et al. Five measurement bases determine pure quantum states on any dimension. Physical Review Letters, 2015, 115 (9): 090401. doi: 10.1103/PhysRevLett.115.090401
    [9]
    Finkelstein J. Pure-state informationally complete and “really” complete measurements. Physical Review A, 2004, 70: 052107. doi: 10.1103/PhysRevA.70.052107
    [10]
    Ma X, Jackson T, Zhou H, et al. Pure-state tomography with the expectation value of Pauli operators. Physical Review A, 2016, 93 (3): 032140. doi: 10.1103/PhysRevA.93.032140
    [11]
    Parashar P, Rana S. N-qubit W states are determined by their bipartite marginals. Physical Review A, 2009, 80: 012319. doi: 10.1103/PhysRevA.80.012319
    [12]
    Torlai G, Mazzola G, Carrasquilla J, et al. Neural-network quantum state tomography. Nature Physics, 2018, 14 (5): 447–450. doi: 10.1038/s41567-018-0048-5
    [13]
    Carleo G, Cirac I, Cranmer K, et al. Machine learning and the physical sciences. Reviews of Modern Physics, 2019, 91 (4): 45002. doi: 10.1103/RevModPhys.91.045002
    [14]
    Goodfellow I J, Bengio Y, Courville A. Deep Learning. Cambridge, MA: MIT Press, 2016.
    [15]
    Carrasquilla J, Torlai G, Melko R G, et al. Reconstructing quantum states with generative models. Nature Machine Intelligence, 2019, 1: 155–161. doi: 10.1038/s42256-019-0028-1
    [16]
    Torlai G. Augmenting quantum mechanics with artificial intelligence. Waterloo, ON, Canada: University of Waterloo, 2018.
    [17]
    Beach M J S, De Vlugt I, Golubeva A, et al. QuCumber: wavefunction reconstruction with neural networks. SciPost Physics, 2019, 7 (1): 009. doi: 10.21468/SciPostPhys.7.1.009
    [18]
    Torlai G, Timar B, van Nieuwenburg E P, et al. Integrating neural networks with a quantum simulator for state reconstruction. Physical Review Letters, 2019, 123 (23): 230504. doi: 10.1103/PhysRevLett.123.230504
    [19]
    Xiao B. Experimental study of quantum entanglement in optical lattices. Hefei: University of Science and Technology of China, 2020.
    [20]
    Yang B, Sun H, Huang C J, et al. Cooling and entangling ultracold atoms in optical lattices. Science, 2020, 369 (6503): 550–553. doi: 10.1126/science.aaz6801
    [21]
    Dai H N, Yang B, Reingruber A, et al. Generation and detection of atomic spin entanglement in optical lattices. Nature Physics, 2016, 12 (8): 783–787. doi: 10.1038/nphys3705
  • 加载中

Catalog

    Figure  1.  Single-qubit flip error mitigation. The target state is the 10-qubit 1-D chain state, which is introduced in Section 3.3. The results (${\text {mean}} \pm {\text {std}}$) are evaluated using Eq. (19) from five independent numerical experiments. The blue points represent the ideal measurement results without errors. The yellow triangles represent the measurement results with flip errors ($ \epsilon=0.01 $), which are simulated by the bit-flips in the randomly chosen data points of the ideal measurement results. The red squares represent the processed measurement results with the RBM model. The green squares represent the processed measurement results with the linear inversion method.

    Figure  2.  Quantum circuit of a 10-qubit 1-D chain state preparation process. The initial state involves the Néel state; the notation U and S stand for ${\sqrt{{\rm{SWAP}}}}^\dagger$ gate and the STO process, respectively.

    Figure  3.  Reconstruction of 1-D chain state with the RBM model. The results (${\text {mean}} \pm {\text {std}}$) are evaluated using the fidelity between the target and reconstructed states with the RBM model in five independent numerical experiments.

    [1]
    Torlai G, Melko R G. Machine-learning quantum states in the NISQ era. Annual Review of Condensed Matter Physics, 2020, 11: 325–344. doi: 10.1146/annurev-conmatphys-031119-050651
    [2]
    Nielsen M A, Chuang I L. Quantum Computation and Quantum Information. Cambridge, UK: Cambridge University Press, 2000.
    [3]
    Prugovečki E. Information-theoretical aspects of quantum measurement. International Journal of Theoretical Physics, 1977, 16 (5): 321–331. doi: 10.1007/BF01807146
    [4]
    Busch P. Informationally complete sets of physical quantities. International Journal of Theoretical Physics, 1991, 30: 1217–1227. doi: 10.1007/BF00671008
    [5]
    Heinosaari T, Mazzarella L, Wolf M M. Quantum tomography under prior information. Communications in Mathematical Physics, 2013, 318 (2): 355–374. doi: 10.1007/s00220-013-1671-8
    [6]
    Chen J, Dawkins H, Ji Z, et al. Uniqueness of quantum states compatible with given measurement results. Physical Review A, 2013, 88: 012109. doi: 10.1103/PhysRevA.88.012109
    [7]
    Flammia S T, Silberfarb A, Caves C M. Minimal informationally complete measurements for pure states. Foundations of Physics, 2005, 35 (12): 1985–2006. doi: 10.1007/s10701-005-8658-z
    [8]
    Goyeneche D, Cañas G, Etcheverry S, et al. Five measurement bases determine pure quantum states on any dimension. Physical Review Letters, 2015, 115 (9): 090401. doi: 10.1103/PhysRevLett.115.090401
    [9]
    Finkelstein J. Pure-state informationally complete and “really” complete measurements. Physical Review A, 2004, 70: 052107. doi: 10.1103/PhysRevA.70.052107
    [10]
    Ma X, Jackson T, Zhou H, et al. Pure-state tomography with the expectation value of Pauli operators. Physical Review A, 2016, 93 (3): 032140. doi: 10.1103/PhysRevA.93.032140
    [11]
    Parashar P, Rana S. N-qubit W states are determined by their bipartite marginals. Physical Review A, 2009, 80: 012319. doi: 10.1103/PhysRevA.80.012319
    [12]
    Torlai G, Mazzola G, Carrasquilla J, et al. Neural-network quantum state tomography. Nature Physics, 2018, 14 (5): 447–450. doi: 10.1038/s41567-018-0048-5
    [13]
    Carleo G, Cirac I, Cranmer K, et al. Machine learning and the physical sciences. Reviews of Modern Physics, 2019, 91 (4): 45002. doi: 10.1103/RevModPhys.91.045002
    [14]
    Goodfellow I J, Bengio Y, Courville A. Deep Learning. Cambridge, MA: MIT Press, 2016.
    [15]
    Carrasquilla J, Torlai G, Melko R G, et al. Reconstructing quantum states with generative models. Nature Machine Intelligence, 2019, 1: 155–161. doi: 10.1038/s42256-019-0028-1
    [16]
    Torlai G. Augmenting quantum mechanics with artificial intelligence. Waterloo, ON, Canada: University of Waterloo, 2018.
    [17]
    Beach M J S, De Vlugt I, Golubeva A, et al. QuCumber: wavefunction reconstruction with neural networks. SciPost Physics, 2019, 7 (1): 009. doi: 10.21468/SciPostPhys.7.1.009
    [18]
    Torlai G, Timar B, van Nieuwenburg E P, et al. Integrating neural networks with a quantum simulator for state reconstruction. Physical Review Letters, 2019, 123 (23): 230504. doi: 10.1103/PhysRevLett.123.230504
    [19]
    Xiao B. Experimental study of quantum entanglement in optical lattices. Hefei: University of Science and Technology of China, 2020.
    [20]
    Yang B, Sun H, Huang C J, et al. Cooling and entangling ultracold atoms in optical lattices. Science, 2020, 369 (6503): 550–553. doi: 10.1126/science.aaz6801
    [21]
    Dai H N, Yang B, Reingruber A, et al. Generation and detection of atomic spin entanglement in optical lattices. Nature Physics, 2016, 12 (8): 783–787. doi: 10.1038/nphys3705

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