[Paper Review] Simulating non-unitary dynamics using quantum signal processing with unitary block encoding

Hans Hon Sang Chan, David Muñoz-Ramo+1 more|arXiv (Cornell University)|Mar 10, 2023

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper introduces a fault-tolerant quantum algorithm using Quantum Eigenvalue Transform with Unitary block-encoding (QET-U) to simulate non-unitary imaginary-time evolution (ITE) on early fault-tolerant quantum computers. By leveraging single-ancilla qubit control and quantum signal processing, it achieves efficient state preparation with success probability scaling quadratically in initial state overlap, enabling ground and thermal state preparation with exponential accuracy in partition function estimation.

ABSTRACT

We adapt a recent advance in resource-frugal quantum signal processing - the Quantum Eigenvalue Transform with Unitary matrices (QET-U) - to explore non-unitary imaginary time evolution on early fault-tolerant quantum computers using exactly emulated quantum circuits. We test strategies for optimising the circuit depth and the probability of successfully preparing the desired imaginary-time evolved states. For the task of ground state preparation, we confirm that the probability of successful post-selection is quadratic in the initial reference state overlap $γ$ as $O(γ^2)$. When applied instead to thermal state preparation, we show QET-U can directly estimate partition functions at exponential cost. Finally, we combine QET-U with Trotter product formula to perform non-normal Hamiltonian simulation in the propagation of Lindbladian open quantum system dynamics. We find that QET-U for non-unitary dynamics is flexible, intuitive and straightforward to use, and suggest ways for delivering quantum advantage in simulation tasks.

Motivation & Objective

• To develop a resource-efficient method for simulating non-unitary imaginary-time evolution on early fault-tolerant quantum computers.
• To overcome the limitations of existing ITE algorithms that require variational optimization or Pauli tomography.
• To demonstrate the feasibility of QET-U for ground state preparation, thermal state simulation, and open quantum system dynamics.
• To analyze and optimize circuit depth and success probability in state preparation via post-selection.

Proposed method

• Adapts QET-U, a recent early fault-tolerant quantum signal processing framework, to simulate non-unitary dynamics via unitary block-encoding.
• Uses controlled unitary evolution with a single ancilla qubit to implement phase rotations based on Chebyshev polynomial expansions.
• Employs symmetric phase factor sequences to approximate exponential decay transforms e−θτ over the Hamiltonian’s eigenvalue spectrum.
• Applies polynomial approximation to map eigenvalues via QSP, enabling transformation of initial states into imaginary-time evolved states.
• Combines QET-U with Trotterized time evolution for Lindbladian dynamics simulation using Pauli gadget decompositions.
• Uses post-selection to project onto the desired evolved state, with success probability dependent on initial state overlap γ.

Experimental results

Research questions

• RQ1 Can QET-U be effectively adapted to simulate non-unitary imaginary-time evolution with minimal ancilla qubits?
• RQ2 What is the scaling of the success probability for state preparation in terms of initial state overlap γ?
• RQ3 How does QET-U compare to variational and QITE methods in terms of classical pre/post-processing and circuit depth?
• RQ4 Can QET-U be used to estimate partition functions directly with exponential accuracy?
• RQ5 Can QET-U be extended to simulate open quantum system dynamics governed by Lindbladian evolution?

Key findings

• The probability of successful post-selection in ground state preparation scales quadratically with the initial state overlap γ, confirming O(γ²) scaling as predicted.
• QET-U enables direct estimation of partition functions at exponential cost, demonstrating utility for finite-temperature statistical mechanics.
• For the 4-site Fermi-Hubbard model, convergence time increases significantly in the strong correlation regime (U/t ≥ 20), where initial state overlap with the true ground state is poor.
• Circuit depth remains prohibitively high for near-term devices, but the framework is suitable for early fault-tolerant architectures.
• The method avoids variational optimization and Pauli tomography, reducing classical computational overhead compared to VITE and QITE.
• QET-U successfully simulates Lindbladian dynamics by combining with Trotter product formula, marking the first demonstration of QSP-based open system simulation.