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Simplification of tensor updates toward performance-complexity balanced quantum computer simulation

Koichi Yanagisawa, Tsuyoshi Okubo, Shota Koshikawa, Tsuyoshi Yoshida, Aruto Hosaka, Synge Todo

TL;DR

Comparisons of the performance and computational cost of SU and CF in quantum circuit simulations show that SU achieves accuracy comparable to CF while reducing computational complexity, indicating that SU provides an efficient alternative for practical quantum circuit simulations.

Abstract

Matrix Product States (MPS) provide a powerful framework for simulating quantum circuits. In practical simulations, tensor updates are typically performed in the canonical form (CF), which corresponds to the Schmidt decomposition and improves approximation accuracy. However, maintaining the canonical form introduces significant computational overhead. An alternative approach, known as the Simple Update (SU), does not enforce the Schmidt decomposition and is expected to reduce computational complexity. In this work, we systematically compare the performance and computational cost of SU and CF in quantum circuit simulations. We benchmark both methods on highly entangled circuits and on a QASM benchmark suite covering a wide range of circuit types. Our results show that SU achieves accuracy comparable to CF while reducing computational complexity, indicating that SU provides an efficient alternative for practical quantum circuit simulations.

Simplification of tensor updates toward performance-complexity balanced quantum computer simulation

TL;DR

Comparisons of the performance and computational cost of SU and CF in quantum circuit simulations show that SU achieves accuracy comparable to CF while reducing computational complexity, indicating that SU provides an efficient alternative for practical quantum circuit simulations.

Abstract

Matrix Product States (MPS) provide a powerful framework for simulating quantum circuits. In practical simulations, tensor updates are typically performed in the canonical form (CF), which corresponds to the Schmidt decomposition and improves approximation accuracy. However, maintaining the canonical form introduces significant computational overhead. An alternative approach, known as the Simple Update (SU), does not enforce the Schmidt decomposition and is expected to reduce computational complexity. In this work, we systematically compare the performance and computational cost of SU and CF in quantum circuit simulations. We benchmark both methods on highly entangled circuits and on a QASM benchmark suite covering a wide range of circuit types. Our results show that SU achieves accuracy comparable to CF while reducing computational complexity, indicating that SU provides an efficient alternative for practical quantum circuit simulations.
Paper Structure (15 sections, 10 equations, 11 figures, 2 tables)

This paper contains 15 sections, 10 equations, 11 figures, 2 tables.

Table of Contents

  1. INTRODUCTION
  2. METHODS
  3. Approximation with MPS
  4. Tensor updates in MPS
  5. One-qubit gate
  6. Two-qubit gate between adjacent qubits
  7. Long-range two-qubit gate
  8. Canonical form
  9. Problems of CF-based quantum computer simulation
  10. Simple update
  11. EVALUATION
  12. Confirming Reduction in Complexity
  13. Investigating Performance
  14. Discussion
  15. CONCLUSION

Figures (11)

  • Figure 1: Tensor network diagrams of (a) SV and (b) MPS, where the nodes and the lines represent tensors and indices, respectively. The connected lines indicate a contraction. In this example, SV is represented by a rank-5 tensor, while the MPS consists of three rank-3 tensors and two rank-2 tensors.
  • Figure 2: The difinition of terms: (a) distant gates, (b) a long-range gate.
  • Figure 3: The procedure of SU: (a) ansatz of the SU, (b) tensors around quantum gate, (c) tensor after contraction, (d) decomposed tensors, (e) approximated tensors, and (f) reverted form of tensors.
  • Figure 4: Exemplified shallow quantum circuit with distant two-qubit gates between adjacent qubits. Though this figure includes only 8 qubits for a small example, we evaluated up to 2000 qubits.
  • Figure 5: Runtime in tensor updates through numerical simulation. Dotted lines are regression lines. SU significantly reduces the runtime for a large number of qubits $N$. Log-scale slopes are around 2 for CF and 1 for SU, respectively. Note that CF refers to the CF-based tensor update. The error bars represent standard errors.
  • ...and 6 more figures