Computing n-time correlation functions without ancilla qubits

TL;DR

This work introduces a method to compute n-time correlation functions using only unitary evolutions on the system of interest, thereby eliminating the need for ancillas and the control operations, and demonstrates the protocol on IBM quantum hardware up to 12 qubits.

Abstract

The $n$-time correlation function is pivotal for establishing connections between theoretical predictions and experimental observations of a quantum system. Conventional methods for computing $n$-time correlation functions on quantum computers, such as the Hadamard test, generally require an ancilla qubit that controls the entire system -- an approach that poses challenges for digital quantum devices with limited qubit connectivity, as well as for analog quantum platforms lacking controlled operations. Here, we introduce a method to compute $n$-time correlation functions using only unitary evolutions on the system of interest, thereby eliminating the need for ancillas and the control operations. This approach substantially relaxes hardware connectivity requirements for digital processors and enables more practical measurements of $n$-time correlation functions on analog platforms. We demonstrate our protocol on IBM quantum hardware up to 12 qubits to measure the single-particle spectrum of the Schwinger model and the out-of-time-order correlator in the transverse-field Ising model. In the demonstration, we further introduce an error mitigation procedure based on signal processing that integrates signal filtering and correlation analysis, and successfully reproduces the noiseless simulation results from the noisy hardware. Our work highlights a route to exploring complex quantum many-body correlation functions in practice, even in the presence of realistic hardware limitations and noise.

Figures (11)

• Figure 1: Schematic of the method. (a) The ancilla-free circuit to measure the $n$-time correlation function (upper panel) and the standard Hadamard test circuit using an ancilla qubit (lower panel). Our approach replaces the control operations in the Hadamard test with the real- or imaginary-time evolution of the operator $O_j$, realized using the red vertical pulses or the blue horizontal pulses, respectively. (b) Computing nested $n$-time commutators and anti-commutators using the parameter-shift rule. $\tau$ is an arbitrary real number of $\mathcal{O}(1)$. Quantum circuits to measure (c) $2$-time commutators and (d) $2$-time anti-commutators. (e) A classical signal-processing method implemented to improve the single-particle spectra by combining signal filtering and correlation analysis. The spectra are derived from the Fourier transformation of the $2$-time correlation functions repeatedly measured on quantum chips.
• Figure 2: Measuring the hadron spectrum of the Schwinger model. (a) Lattice and hadron operator of the Schwinger model. (b) Comparison of the measurement circuits for the hadron spectrum using the ancilla-free circuit (left panel) and Hadamard test (right panel), and their physical realization on digital quantum chips with linear qubit connectivity (dashed boxes). (c) Measurement results of 2-time correlation function $C^b_{k,0}(t)$ on versus time $t$ with four different momentum $k$. The grey solid and dashed lines denote exact diagonalization and noiseless Trotter results. We perform the measurement twice for correlation analysis (CA), labeled by raw 1 and raw 2. (d) Normalized hadron spectrum of the two raw results, classically processed using CA, CA and filtering, and the noiseless Trotter spectrum. The highest energy peak of each $k$ is marked by a red dot, and fitted by the red dashed line using the hadron dispersion relation. (e) Effect of the classical signal-processing method. The error (upper panel) is the position of energy peaks between the measured and the noiseless Trotter results. $m_h c^2$ (lower panel) is obtained from the fitted hadron dispersion relation, with the error bars denoting the fitting errors. The grey dashed line denotes the center value of the noiseless Trotter result.
• Figure 3: Measuring OTOC of the transverse-field Ising model (TIM). (a) The ancilla-free circuits to measure the imaginary part (with the red gate $e^{-i\tau V}$) and the real part (with the blue gate $e^{-\tau V}$) of OTOC. (b) Lattice of TIM and the operator in OTOC. (c) The real and imaginary parts of OTOC $F(t)$ measured on versus the time $t$. The grey solid and grey dashed lines denote exact diagonalization and noiseless Trotter results, respectively.
• Figure 4: Quantum circuits to measure the 3-time nested anti-commutator $\bra{\phi}[[O_2(t_2),O_1(t_1)]_+,O_0(t_0)]_+\ket{\phi}$ and its correction factor.
• Figure S5: Measurement circuits for $3$-time nested commutator (upper panel) and its bang-bang version suitable for digital-analog platforms (lower panel). $H$ is the native Hamiltonian of the analog platform. $\Lambda$ is a rescaling factor. The evolution time of each block is labeled above each block. Compared with the original circuit, the bang-bang circuit has the system evolution on during the whole time.