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Identity Check Problem for Shallow Quantum Circuits

Authors Sergey Bravyi , Natalie Parham , Minh Tran

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LIPIcs.ITCS.2026.27.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Quantum complexity theory
Keywords
  • Quantum computing
  • Identity check problem
  • quantum circuits
  • classical simulation of quantum computation
  • shallow circuits

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Abstract

Verifying that a quantum circuit correctly implements a desired transformation is essential for validating quantum algorithms. We consider the closely related identity check problem: given a quantum circuit U, estimate the diamond-norm distance between U and the identity channel. Ji and Wu showed that estimating this distance to within an additive 1/poly error is QMA-hard, even when U is constant-depth and 1D local - ruling out efficient algorithms in this regime.
We show that this hardness barrier disappears if one seeks a constant multiplicative-approximation instead. We present a classical algorithm that, for shallow geometrically local D-dimensional circuits, approximates the distance to the identity within a factor α = D+1, provided that the circuit is sufficiently close to the identity. The runtime of the algorithm scales linearly with the number of qubits for any constant circuit depth and spatial dimension.
We also show that the operator-norm distance to the identity ‖U-I‖ can be efficiently approximated within a factor α = 5 for shallow 1D circuits and, under a certain technical condition, within a factor α = 2D+3 for shallow D-dimensional circuits. A numerical implementation of the identity check algorithm is reported for 1D Trotter circuits with up to 100 qubits.

Cite As Get BibTex

Sergey Bravyi, Natalie Parham, and Minh Tran. Identity Check Problem for Shallow Quantum Circuits. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.27

Author Details

Sergey Bravyi
  • IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, USA
Natalie Parham
  • Columbia University, USA
Minh Tran
  • IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, USA

Funding

  • Parham, Natalie: NP is supported by AFOSR award FA9550-21-1-0040, NSF CAREER award CCF-2144219, and the Sloan Foundation.

Acknowledgements

SB thanks Steven Flammia and Kristan Temme for helpful discussions. MCT thanks Kunal Sharma for helpful discussions. This work was partially completed while NP was interning at IBM Quantum.

References

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