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Efficient Circuit-Based Quantum State Tomography via Sparse Entry Optimization

Chi-Kwong Li, Kevin Yipu Wu, Zherui Zhang

TL;DR

The paper tackles scalable quantum state tomography for pure states that are sparse in the computational basis, introducing a circuit-based framework that leverages the sparsity pattern to bound resources by the state’s intrinsic geometry. It builds a weighted support graph on the $k$ nonzero amplitudes, uses a minimum spanning tree to organize $k-1$ edge-resolutions, and provides two circuit implementations for edge resolution: entanglement-assisted (ENT) with CNOTs and entanglement-free partial-mixing (PM). The authors derive concrete resource guarantees—$M = O(k)$ measurement settings, $G_{\mathrm{CNOT}} = O(nk)$ two-qubit gates in the worst case, and $S = O(k/\epsilon^2)$ samples—along with a regression-ready classical post-processing plan, and extend the approach to closed-system process tomography. Numerical simulations with realistic noise (IBM Marrakesh) validate the framework across various sparse-state regimes, showing strong performance of ENT for larger edge weights and PM for small distances, with process tomography demonstrated as well. Overall, the method offers a structure-aware, near-term-friendly path to efficient tomography that scales with sparsity $k$ rather than the full Hilbert-space dimension $2^n$, enabling practical verification of large quantum systems.

Abstract

Many quantum states arising in algorithms and physical systems occupy only a small, structured subset of the exponentially large Hilbert space, yet standard quantum state tomography fails to exploit this structure. We present an efficient circuit-based tomography framework for pure quantum states that are sparse in a computational basis. For an $n$-qubit state supported on $k$ basis elements, the protocol reconstructs all amplitudes using $1 + 2(k-1)$ measurement settings. The method admits both entanglement-assisted and entanglement-free implementations, enabling explicit tradeoffs between two-qubit gate usage and statistical noise. We derive bounds on the required number of CNOT gates from the combinatorial structure of the state support and analyze their effect on reconstruction infidelity. The framework extends naturally to closed-system process tomography and is validated via numerical simulations using Qiskit.

Efficient Circuit-Based Quantum State Tomography via Sparse Entry Optimization

TL;DR

The paper tackles scalable quantum state tomography for pure states that are sparse in the computational basis, introducing a circuit-based framework that leverages the sparsity pattern to bound resources by the state’s intrinsic geometry. It builds a weighted support graph on the nonzero amplitudes, uses a minimum spanning tree to organize edge-resolutions, and provides two circuit implementations for edge resolution: entanglement-assisted (ENT) with CNOTs and entanglement-free partial-mixing (PM). The authors derive concrete resource guarantees— measurement settings, two-qubit gates in the worst case, and samples—along with a regression-ready classical post-processing plan, and extend the approach to closed-system process tomography. Numerical simulations with realistic noise (IBM Marrakesh) validate the framework across various sparse-state regimes, showing strong performance of ENT for larger edge weights and PM for small distances, with process tomography demonstrated as well. Overall, the method offers a structure-aware, near-term-friendly path to efficient tomography that scales with sparsity rather than the full Hilbert-space dimension , enabling practical verification of large quantum systems.

Abstract

Many quantum states arising in algorithms and physical systems occupy only a small, structured subset of the exponentially large Hilbert space, yet standard quantum state tomography fails to exploit this structure. We present an efficient circuit-based tomography framework for pure quantum states that are sparse in a computational basis. For an -qubit state supported on basis elements, the protocol reconstructs all amplitudes using measurement settings. The method admits both entanglement-assisted and entanglement-free implementations, enabling explicit tradeoffs between two-qubit gate usage and statistical noise. We derive bounds on the required number of CNOT gates from the combinatorial structure of the state support and analyze their effect on reconstruction infidelity. The framework extends naturally to closed-system process tomography and is validated via numerical simulations using Qiskit.
Paper Structure (28 sections, 5 theorems, 26 equations, 4 figures, 1 table)

This paper contains 28 sections, 5 theorems, 26 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Algorithm
  4. Full Support States
  5. Two-term superpositions
  6. Partial mixing tomography
  7. Sparse Tomography Algorithm using a Graph Theory
  8. Structural Results and Edge-Resolution Guarantees
  9. Complexity Guarantees
  10. Measurement Settings
  11. CNOT gate complexity
  12. Sample Complexity
  13. Classical Post-Processing Complexity
  14. Regime Selection
  15. Noise mechanisms in the entangling and partial-mixing regimes
  16. ...and 13 more sections

Key Result

Proposition 3.1

Suppose $|\psi{\rangle}$ is $n$-qubit state with no zero entries. Then $|\psi{\rangle}$ can be reconstructed using the measurements of $|\psi{\rangle}, H_{n-1}|\psi{\rangle}, \cdots H_0|\psi{\rangle}, V_{n-1}|\psi{\rangle}, \dots, V_0|\psi{\rangle}$.

Figures (4)

  • Figure 1: Comparison of entanglement-assisted (ENT) and partial-mixing (PM) tomography across different support sizes. Each subfigure shows the minimum-spanning-tree structure of the nonzero amplitudes alongside the corresponding infidelity distributions.
  • Figure 2: log-infidelity distributions for GHZ chains of size $k=2$--$10$ under the IBM Marrakesh noise model, comparing entangling (blue) and partial-mixing (orange) protocols.
  • Figure 3: Noise–parameter sweep for a GHZ-6 state. Each panel shows reconstruction log-infidelity as a function of the two-qubit depolarizing rate $p_{2q}$ and measurement error $p_{\mathrm{meas}}$. The heatmaps illustrate the sensitivity of the entangling and partial-mixing tomography protocols to gate and readout noise, revealing the parameter regimes in which each method breaks down.
  • Figure 4: Two-qubit process tomography (see Sec. \ref{['ssec:process_tomography']}).

Theorems & Definitions (6)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Proposition 3.3
  • Theorem 4.1
  • Proposition 4.2