Quantum Phase Estimation by Compressed Sensing

TL;DR

This paper introduces a compressed-sensing approach to quantum phase estimation (QPE) suitable for early fault-tolerant quantum devices. By modeling the QEEP signal as a sparse combination of energy eigenvalues and employing a grid-shift parameter to handle off-grid components, the method achieves Heisenberg-limited scaling with discrete, sparse time samples. The authors provide a rigorous main theorem guaranteeing recovery accuracy and robustness under realistic noise, along with detailed numerical comparisons to existing non-adaptive QPE methods. Overall, the work advances practical QPE by reducing sampling requirements while maintaining high accuracy and resilience to noise, with implications for quantum simulation and chemistry.

Abstract

As a signal recovery algorithm, compressed sensing is particularly useful when the data has low-complexity and samples are rare, which matches perfectly with the task of quantum phase estimation (QPE). In this work we present a new Heisenberg-limited QPE algorithm for early quantum computers based on compressed sensing. More specifically, given many copies of a proper initial state and queries to some unitary operators, our algorithm is able to recover the frequency with a total runtime $\mathcal{O}(ε^{-1}\text{poly}\log(ε^{-1}))$, where $ε$ is the accuracy. Moreover, the maximal runtime satisfies $T_{\max}ε\ll π$, which is comparable to the state of art algorithms, and our algorithm is also robust against certain amount of noise from sampling. We also consider the more general quantum eigenvalue estimation problem (QEEP) and show numerically that the off-grid compressed sensing can be a strong candidate for solving the QEEP.

Figures (3)

• Figure 1: The one-ancilla quantum circuit used in Kitaev-type QPE algorithms. The final measurement is done in the $Z$ basis. In terms of the measurement outcome, we regard the $|0\rangle$ state as obtaining value $+1$, and the $|1\rangle$ state as obtaining value $-1$. $\mathbf{H}$ is the Hadamard gate; $\mathbf{W}$ has two choices: when $\mathbf{W} = I$, the measurement outcome is $\pm 1$ with probability $(1 \pm \text{Re}(\langle\Phi|U(t)|\Phi\rangle))/2$ respectively. When $\mathbf{W} = \mathbb{S}^{\dagger}$, the complex conjugation of the phase gate, the measurement outcome is $\pm 1$ with probability $(1 \pm \text{Im}(\langle\Phi|U(t)|\Phi\rangle))/2$ instead. After averaging over many test outcomes, we obtain an estimate of the true signal $\langle\Phi|U(t)|\Phi\rangle$.
• Figure 2: Comparison between the setups of the four algorithms. Here we set $T_n = \lfloor 100 \times (1.4)^{n}\rfloor, n = 1,2,3,4,5$. In Algorithm \ref{['algorithm:single_eigenvalue']}, we set $S = 1, r = 2.3\ln T_n/T_n,\tau = 1, \sigma = 0.2\sqrt{2.3\ln T_n}, J = 100, M_{\mathrm{H}} = 100$ and let $\mathcal{K} = \{\arg\max s_{\nu_\ast}\}$. In ML-QCELS, we set $N = 8, N_s = 50, J = \lfloor 4\ln T_n\rfloor$ (the meanings of the parameters can be found in PRXQuantum.4.020331). In MM-QCELS, we set $K = 2, N_T = 30, \gamma = 1, N_s = 100, J = \lfloor 4\ln T_n\rfloor$ (the meanings of the parameters can be found in ding2023simultaneous). In QMEGS, we set $K = 10, \mathrm{dx} = 10^{-4}, \alpha = 5, N = 10 + 2\lfloor\ln (2T_n)\rfloor$ (the meanings of the parameters can be found in ding2024quantum). The codes are available online github.
• Figure 3: Comparison between Algorithm \ref{['algorithm:single_eigenvalue']}, ML-QCELS, MM-QCELS and QMEGS with respect to signals generated by Ising model (Eq. (\ref{['equ:ising']}), left column) or Hubbard model (Eq. (\ref{['equ:hubbard']}), right column). Subfigures (a),(b) represent $\alpha = 1/8$ ($p_0 \approx 0.88$); (c),(d) represent $\alpha = 1/4$ ($p_0 \approx 0.75$); (e),(f) represent $\alpha = 1/2$ ($p_0 \approx 0.5$).

Theorems & Definitions (25)

• Theorem 1: Informal
• Lemma 1
• Lemma 2
• Theorem 2
• Remark
• Theorem 3
• proof
• Theorem 4
• Lemma 3
• Lemma 4: A good $\nu$ generates a good solution