Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum algorithm for dephasing of coupled systems: decoupling and IQP duality

Sabrina Yue Wang, Raul A. Santos

TL;DR

This work addresses the challenge of simulating open quantum systems described by Lindbladian dynamics, focusing on unital generators and proposing a resource-efficient quantum algorithm that approximates evolution by sampling mixed-unitary channels via unitary circuits, with error $O(t^2)$ for short times. It extends to general Lindbladians by incorporating ancillas, and introduces a decoupling (Dcube) scheme for interacting dephasing Lindbladians that reduces bipartite dynamics to conditioned unitary evolutions on one subsystem driven by the other. A key advance is tracing bosonic degrees of freedom in electron-phonon models to yield an effective IQP-circuit description for the fermionic sector, revealing a concrete link between dissipative dynamics and circuit sampling, with potential classical hardness via IQP circuit sampling. Numerically, the framework is validated on a spinless fermion dimer and a spinful Fermi-Hubbard dimer, highlighting both the promise and practical challenges (e.g., boson-space truncation) of implementing these dissipative simulations on near-term hardware. The results illuminate how dissipation, disorder, and non-Markovian effects can be captured within a programmable quantum circuit paradigm and point to fundamental questions about the classical simulability of such quantum-dissipative processes.

Abstract

Noise and decoherence are ubiquitous in the dynamics of quantum systems coupled to an external environment. In the regime where environmental correlations decay rapidly, the evolution of a subsytem is well described by a Lindblad quantum master equation. In this work, we introduce a quantum algorithm for simulating unital Lindbladian dynamics by sampling unitary quantum channels without extra ancillas. Using ancillary qubits we show that this algorithm allows approximating general Lindbladians as well. For interacting dephasing Lindbladians coupling two subsystems, we develop a decoupling scheme that reduces the circuit complexity of the simulation. This is achieved by sampling from a time-correlated probability distribution - determined by the evolution of one subsystem, which specifies the stochastic circuit implemented on the complementary subsystem. We demonstrate our approach by studying a model of bosons coupled to fermions via dephasing, which naturally arises from anharmonic effects in an electron-phonon system coupled to a bath. Our method enables tracing out the bosonic degrees of freedom, reducing part of the dynamics to sampling an instantaneous quantum polynomial (IQP) circuit. The sampled bitstrings then define a corresponding fermionic problem, which in the non-interacting case can be solved efficiently classically. We comment on the computational complexity of this class of dissipative problems, using the known fact that sampling from IQP circuits is believed to be difficult classically.

Quantum algorithm for dephasing of coupled systems: decoupling and IQP duality

TL;DR

This work addresses the challenge of simulating open quantum systems described by Lindbladian dynamics, focusing on unital generators and proposing a resource-efficient quantum algorithm that approximates evolution by sampling mixed-unitary channels via unitary circuits, with error for short times. It extends to general Lindbladians by incorporating ancillas, and introduces a decoupling (Dcube) scheme for interacting dephasing Lindbladians that reduces bipartite dynamics to conditioned unitary evolutions on one subsystem driven by the other. A key advance is tracing bosonic degrees of freedom in electron-phonon models to yield an effective IQP-circuit description for the fermionic sector, revealing a concrete link between dissipative dynamics and circuit sampling, with potential classical hardness via IQP circuit sampling. Numerically, the framework is validated on a spinless fermion dimer and a spinful Fermi-Hubbard dimer, highlighting both the promise and practical challenges (e.g., boson-space truncation) of implementing these dissipative simulations on near-term hardware. The results illuminate how dissipation, disorder, and non-Markovian effects can be captured within a programmable quantum circuit paradigm and point to fundamental questions about the classical simulability of such quantum-dissipative processes.

Abstract

Noise and decoherence are ubiquitous in the dynamics of quantum systems coupled to an external environment. In the regime where environmental correlations decay rapidly, the evolution of a subsytem is well described by a Lindblad quantum master equation. In this work, we introduce a quantum algorithm for simulating unital Lindbladian dynamics by sampling unitary quantum channels without extra ancillas. Using ancillary qubits we show that this algorithm allows approximating general Lindbladians as well. For interacting dephasing Lindbladians coupling two subsystems, we develop a decoupling scheme that reduces the circuit complexity of the simulation. This is achieved by sampling from a time-correlated probability distribution - determined by the evolution of one subsystem, which specifies the stochastic circuit implemented on the complementary subsystem. We demonstrate our approach by studying a model of bosons coupled to fermions via dephasing, which naturally arises from anharmonic effects in an electron-phonon system coupled to a bath. Our method enables tracing out the bosonic degrees of freedom, reducing part of the dynamics to sampling an instantaneous quantum polynomial (IQP) circuit. The sampled bitstrings then define a corresponding fermionic problem, which in the non-interacting case can be solved efficiently classically. We comment on the computational complexity of this class of dissipative problems, using the known fact that sampling from IQP circuits is believed to be difficult classically.
Paper Structure (20 sections, 4 theorems, 71 equations, 7 figures, 2 algorithms)

This paper contains 20 sections, 4 theorems, 71 equations, 7 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Relation to previous work
  3. Main results
  4. Motivation
  5. Extension to general Lindbladians
  6. Applications
  7. Connection with simulation of disordered systems
  8. Electron-phonon systems
  9. Circuit construction of auxiliary dissipative Lindbladian
  10. Nontrivial steady states of $\mathcal{L}_{\rm IQP}$
  11. Numerical results
  12. Free fermion dimer coupled to bosons
  13. Dependence on Hilbert space truncation
  14. Behaviour of auxiliary dissipative Lindbladian $\Gamma$ distribution
  15. Green's function of the spinful Fermi-Hubbard dimer under various dephasing
  16. ...and 5 more sections

Key Result

Theorem 1

The time evolution with the unital Lindblad operator where $N_L$ is the number of jump operators in the dissipative component and $L_i$ is hermitian, can be approximated by the expectation value of the stochastic channel where each $s_j=\pm 1$ in ${\bm s}=(s_1,\dots s_{N_L})$ is a uniformly distributed random variable with probability distribution $p_{\bm s}=\frac{1}{2^{N_L}}$. The error incurre

Figures (7)

  • Figure 1: Decoupling gadget: The average of a channel with interaction term $e^{is\sqrt{t}AB}$ where $s=\{-1,1\}$ can be expressed as the outcome of a different channel where one subsystem is coupled to an ancilla through the Pauli Z operator and then the ancilla is measured. The result of that measurement defines the circuit that is implemented in the other subsystem.
  • Figure 2: Schematic of the dimer in the presence of a bath and quantum Brownian particles on each site. Physical sites are denoted by the black circles with site index $j$. The harmonic oscillator with energy level spacing $\omega_j$ denote the Brownian particle on each site, relating to the Hamiltonian $H_Q$. The gray oval encompassing the entire site represents the boson bath with a continuum of energy levels $\tilde{\omega}_{j,\alpha}$ (see also \ref{['appendix:epi_lindbladian_derivation']}). On each site, the bath, Brownian particle and fermions are all coupled together with interaction strength $g_j$. Fermions can hop between the sites with hopping strength $J$ and are described by the Hamiltonian $H_F$.
  • Figure 3: Density time dynamics of the spinless free fermion dimer induced by the dephasing Lindbladian of \ref{['eq:Lind_eph']} compares (a) 'exact Lindbladian' against 'circuit' method with the fermion density absolute error in (b). The dashed lines are linear fits to the logarithm of the absolute density error with the slopes in the respective legend according to $N$ Trotter steps. The number of ancillas to realise the auxiliary dissipative Lindbladian in \ref{['eqn:DIQP']} is always set equal to the total number of Trotter steps, $N_{\eta}=N$. Each application of this map uses $N_{\rho}=\lceil {N}^{\frac{1}{4}}\rceil$ internal Trotter steps. A total of $10$k samples are taken per experiment across all different Trotter steps for the 'circuit' method. Similarly, the time dynamics of (c) 'exact Lindbladian' against 'stochastic' method with absolute fermion density error are compared in (d). $5$k samples of uniformly sampled stochastic channels are used across all $N$ Trotter steps. $N_b=8$ is set for 'exact Lindbladian' and 'stochastic' simulations. The initial state is a Fock state with a single fermion on site one and $g=4$ across all simulations.
  • Figure 4: (LHS) Time dynamics of particle density of a spinless Fermi-Hubbard dimer under the Lindbladian \ref{['eq:Lind_eph']}. $\langle n^d_{i} \rangle$ represents the number density of site $i \in [0,1]$ and particle identity $d \in [f,b]$ (f=fermion, b=boson), for two instances of $N_b$. The initial state is a single fermion on site $i=0$. Each plot is associated with the boson truncation level of maximum $N_b-1$ bosons per site. (RHS) The maximum change in fermion density across all sites with increasing boson Hilbert space dimension in the 'exact Lindbladian' simulations for the dimer at $g=4$.
  • Figure 5: The full decomposition of the probability distribution of the dimer $\Gamma(t)$ for different bitstring $\bm{\gamma}$s at various fixed total evolution time. $g=4$, $\omega=J=1$, $N_{\eta}=5$ and $N_{\rho}=2$ and $N=2$ first order Trotter steps. The 'exact' method refer to exactly computing $\Gamma(t)$. The 'circuit' method refers to sampling from the corresponding noiseless circuit that approximates $\Gamma(t)$. $N_s=10$k samples were drawn and the error bars signify the sampling error $\frac{1}{\sqrt{N_s}}$.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Theorem 1: Dissipative Dynamics - simplified
  • Theorem 2: Dissipative Decoupled Dynamics: Dcube
  • Lemma 3: Decoupling lemma
  • Lemma 4
  • proof