Spectral functions on a quantum computer through system-environment interaction

Abstract

Spectral functions measured with angle-resolved photoemission spectroscopy (ARPES) provide key insight to elucidate the band structure of materials. Comparison with theory requires computing dynamical one-point functions in some equilibrium state, which can be classically challenging. Their measurement on quantum computers poses multiple problems and comes with a large sampling overhead when standard techniques are used. We introduce an efficient way of measuring spectral functions on a quantum computer by directly modeling the interaction of the system with the environment involved in ARPES experiments. We develop quantum circuits whose local expectation values are proportional to the spectral function $A(k,ω)$ for all momentum $k$ and a specific chosen frequency $ω$. Although coming with a qubit and two-qubit gate overhead, our approach requires $O(N)$ times less sampling than previous approaches, translating into a factor $O(N)$ faster in runtime, and is particularly adapted to ion-trap quantum computers. The algorithm requires to implement a fermionic Fourier transform (FFT). We write out an efficient gate decomposition for generic radix-$n$ FFT and benchmark it on hardware for radix-$3$ on $27$ qubits. We finally demonstrate our algorithm on a Quantinuum System Model H2 ion-trap system, computing the spectral function on a one-dimensional system of $27$ sites, using $54$ qubits.

The paper addresses the measurement cost of fermionic spectral functions on quantum computers, where real-space dynamical-correlation protocols require separate measurements of noncommuting fermionic operators. It proposes coupling an $N$-mode, particle-conserving system to an $N$-mode fermionic environment with tunable on-site energy $\omega$ and hopping strength $\epsilon/2$. After evolution for time $t$, a fermionic Fourier transform and computational-basis readout measure all environment momentum occupations simultaneously. A perturbative derivation gives $\langle n(k)\rangle=\epsilon^2(A^+\star\hat\varphi)(k,\omega)+O(\epsilon^4)$, with $\hat\varphi(\omega)=\sin^2(\omega t/2)/\omega^2$. Empty and filled environments access the removal and addition contributions, respectively. Relative to the real-space correlation approach considered, the protocol claims an $O(N)$ sampling and runtime reduction, at the cost of doubling the qubits, adding $2N$ two-qubit gates per Trotter step, and performing a Fourier transform. Generic-radix fermionic Fourier-transform circuits and optimized fermionic permutation networks are also developed. Hardware tests benchmark radix-3 transforms on 27 qubits and measure the spectrum of a 27-site free-fermion chain using 54 qubits, 10 Trotter steps, $t=5$, $\epsilon=0.5$, and 2,172 two-qubit gates per circuit. Raw measurements reproduce the band shape with reduced contrast and a noise floor. Finite evolution time broadens frequency features by roughly $1/t$. Weak-coupling extrapolation amplifies sampling costs; finite coupling instead introduces bias and possible ghost bands. Its effects are analyzed exactly for free fermions, with $\epsilon t\lesssim\pi$ suggested as a practical regime rather than established generally for interacting systems. The hardware demonstration remains classically simulable.