Hardware-efficient quantum phase estimation via local control

TL;DR

The paper tackles the challenge of performing quantum phase estimation on near-term devices by replacing globally controlled evolutions with three local-control schemes that extract the phase of the Loschmidt echo, $g(t)=\langle\psi|U(t)|\psi\rangle = r(t) e^{i\phi(t)}$, thereby dramatically reducing circuit depth. It introduces the Sequential Hadamard test, Direct phase gradient, and ITE phase gradient methods, with rigorous additive-error sampling bounds and explicit scaling in system size, evolution time, and accuracy. A numerical study on a non-integrable Ising model demonstrates the practical viability and noise robustness of the approaches, including LDOS reconstruction under hardware noise. The work provides a practical pathway for spectroscopy-like measurements in large many-body quantum systems on current hardware, applicable to both digital and analog-digital platforms without requiring reference states.

Abstract

Quantum phase estimation plays a central role in quantum simulation as it enables the study of spectral properties of many-body quantum systems. Most variants of the phase estimation algorithm require the application of the global unitary evolution conditioned on the state of one or more auxiliary qubits, posing a significant challenge for current quantum devices. In this work, we present an approach to quantum phase estimation that uses only locally controlled operations, resulting in a significantly reduced circuit depth. At the heart of our approach are efficient routines to measure the complex phase of the expectation value of the time-evolution operator, the so-called Loschmidt echo, for both circuit dynamics and Hamiltonian dynamics. By tracking changes in the phase during the dynamics, the routines trade circuit depth for an increased sampling cost and classical postprocessing. Our approach does not rely on reference states and is applicable to any efficiently preparable state, regardless of its correlations. We provide a comprehensive analysis of the sample complexity and illustrate the results with numerical simulations. Our methods offer a practical pathway for measuring spectral properties in large many-body quantum systems using current quantum devices.

Figures (10)

• Figure 1: (a) The complex Loschmidt echo quantifies the overlap of an initial state $\ket{\psi}$ with the state obtained after a unitary evolution. The evolution can be generated by a Hamiltonian $H$, as illustrated here, or by a quantum circuit. (b) While the standard Hadamard test requires global control to measure $g(t)$, our methods extract its phase $\phi(t)$ using only local control, thereby reducing circuit depth at the expense of increased sampling. (c) By applying a Fourier transform to the time series of the Loschmidt echo, we obtain a broadened local density of states, shown in front of gray spectral lines.
• Figure 2: Overview of three methods to obtain the complex Loschmidt echo without global control. (Left) The sequential control method relies on implementing as many different circuits as there are gates, where a single gate is controlled in each circuit. It works for arbitrary circuits and is well suited for digital devices. (Center) The direct phase gradient method applies to Hamiltonian dynamics, which makes it compatible with hybrid analog-digital devices. The method separately computes the derivative of the phase $\mathrm{d}\phi(t)/\mathrm{d} t$ for each local Pauli term $P_j$ of a Hamiltonian $H=\sum_jh_j=\sum_j\lambda_jP_j$. (Right) Alternatively, we can obtain the phase gradient by applying imaginary time evolution (ITE) to an arbitrary initial state using auxiliary qubits. The ITE is realized for each Hamiltonian term $h_j$ separately, using a single auxiliary qubit per term.
• Figure 3: Circuit diagram for the $L^\text{th}$ iteration of the sequential Hadamard test.
• Figure 4: Circuit diagram for the direct phase gradient method. The measurement yields the expression in Eq. \ref{['eq:direct_phase_circuit']}.
• Figure 5: Circuit used to apply the local ITE operator $\propto\exp(\pm P_j\lambda_j\tau)$ to an initial quantum state $\ket{\psi}$ using $R_y=\exp(-i\theta\sigma^y/2)$ rotations on the single auxiliary qubit, where $\theta$ depends on $\lambda_j$ (see main text). The ITE is successfully applied when the auxiliary qubit is measured to be in the $\ket{0}$-state at the end of the circuit.