Tensor-based phase difference estimation on time series analysis

TL;DR

This work introduces a tensor-network–based time-series framework for estimating energy gaps with a QPE-type algorithm by compressing both time-evolution and state-preparation circuits into MPS/MPO representations, enabling shallow, nearest-neighbor circuits. It leverages four-circuit measurements to extract complex time-series data $s_t$ and employs classical processing (matrix pencil/direct fitting) to recover gaps, notably $\Delta_{(0,1)} = E_1 - E_0$, from spectral components. To boost accuracy on realistic hardware, the authors develop algorithmic error mitigation (AEM) and an iterative overlap-enhancement procedure for state preparation, demonstrating improved gap estimation on 8-, 36-, and 52-qubit Hubbard models on simulators and IBM devices. The results illustrate a viable pathway toward practical near-term quantum applications and pave the way for error-corrected quantum devices by combining tensor-network compression with refined classical post-processing and hardware-aware mitigation.

Abstract

We propose a phase-difference estimation algorithm based on the tensor-network circuit compression, leveraging time-evolution data to pursue scalability and higher accuracy on a quantum phase estimation (QPE)-type algorithm. Using tensor networks, we construct circuits composed solely of nearest-neighbor gates and extract time-evolution data by four-type circuit measurements. In addition, to enhance the accuracy of time-evolution and state-preparation circuits, we propose techniques based on algorithmic error mitigation and on iterative circuit optimization combined with merging into matrix product states, respectively. Verifications using a noiseless simulator for the 8-qubit one-dimensional Hubbard model using an ancilla qubit show that the proposed algorithm achieves accuracies with 0.4--4.7\% error from a true energy gap on an appropriate time-step size, and that accuracy improvements due to the algorithmic error mitigation are observed. We also confirm the enhancement of the overlap with matrix product states through iterative optimization. Finally, the proposed algorithm is demonstrated on IBM Heron devices with Q-CTRL error suppression for 8-, 36-, and 52-qubit models using more than 5,000 2-qubit gates. These largest-scale demonstrations for the QPE-type algorithm represent significant progress not only toward practical applications of near-term quantum computing but also toward preparation for the era of error-corrected quantum devices.

Figures (10)

• Figure 1: Overview of our proposal. $P(\theta) = 100e^{i\theta}$ represents the phase gate with the parameter $\theta$ taking $0, \pi/2, \pi,$ and $3\pi/2$. $s_t$ is a signal to construct a time series data $\{(t,s_t )\}_{t=t_1}^{t_\mathrm{max}}$, where $t_1$ and $t_\mathrm{max}$ are $\Delta t$ and a maximum time, respectively.
• Figure 2: Summary of the unitary compression in our previous work. (a) Initial preparations for $U_\mathrm{evol}$ and $U_\mathrm{ref}^{\dagger}$. (b) Gate optimization for a target gate. (c) Optimization sequence for gates. (d) Substitution for $U_\mathrm{prep}$.
• Figure 3: Calculations for $M_{ij}(t)$ and $L_{i}(t)$. (a) Transformation from circuits of $\ket{\psi_\mathrm{prep}}$ and $U_\mathrm{evol}^i$ to the MPS and MPO, respectively. (b) Calculation procedure for $M_{ij}$. Note that $L_i$ can also be calculated in almost the same way.
• Figure 4: Procedure of the overlap enhancement. (a) Circuit optimization. (b) Circuit merging. (c) Deep circuit implementation.
• Figure 5: Energy gap errors, $\Delta_{(0,1)}^\mathrm{est} - \Delta_{(0,1)}^\mathrm{ref}$, for time step with several step sizes $\Delta t$.