[Paper Review] Quantum circuits for solving local fermion-to-qubit mappings

Jannes Nys, Giuseppe Carleo|arXiv (Cornell University)|Aug 15, 2022

Quantum and electron transport phenomena4 citations

TL;DR

This paper introduces quantum circuits that exactly enforce Gauss-law constraints in local fermion-to-qubit mappings, enabling fully local time-evolution circuits with constant depth per Trotter step. The method uses auxiliary fermionic degrees of freedom and unitary operations to maintain locality and fermionic statistics, significantly improving scalability for simulating fermionic systems in d>1 dimensions on NISQ devices.

ABSTRACT

Local Hamiltonians of fermionic systems on a lattice can be mapped onto local qubit Hamiltonians. Maintaining the locality of the operators comes at the expense of increasing the Hilbert space with auxiliary degrees of freedom. In order to retrieve the lower-dimensional physical Hilbert space that represents fermionic degrees of freedom, one must satisfy a set of constraints. In this work, we introduce quantum circuits that exactly satisfy these stringent constraints. We demonstrate how maintaining locality allows one to carry out a Trotterized time-evolution with constant circuit depth per time step. Our construction is particularly advantageous to simulate the time evolution operator of fermionic systems in d>1 dimensions. We also discuss how these families of circuits can be used as variational quantum states, focusing on two approaches: a first one based on general constant-fermion-number gates, and a second one based on the Hamiltonian variational ansatz where the eigenstates are represented by parametrized time-evolution operators. We apply our methods to the problem of finding the ground state and time-evolved states of the $t$-$V$ model.

Motivation & Objective

• To address the non-locality of Jordan-Wigner strings in fermionic quantum simulations on digital quantum computers.
• To develop quantum circuits that exactly satisfy the Gauss-law constraints arising from local fermion-to-qubit mappings with auxiliary degrees of freedom.
• To enable constant-depth Trotterized time evolution for local fermionic Hamiltonians in higher dimensions (d>1).
• To construct variational quantum circuits preserving fermion number and periodicity constraints for ground state and time-evolved state preparation.
• To provide a scalable, hardware-efficient framework for simulating local fermionic systems beyond traditional Jordan-Wigner encodings.

Proposed method

• The method employs a local fermion-to-qubit mapping that introduces auxiliary fermionic modes to eliminate non-local Jordan-Wigner strings.
• It constructs quantum circuits using local unitary operations to exactly enforce Gauss-law constraints, ensuring the physical Hilbert space is preserved.
• The time-evolution circuit uses only local gates, achieving constant circuit depth per Trotter step, independent of system size.
• The approach leverages a generalized Jordan-Wigner transformation with auxiliary fermions that store parity information locally.
• Variational circuits are designed using constant-fermion-number gates and parametrized time-evolution operators based on the Hamiltonian variational ansatz.
• The method is applicable to arbitrary lattice topologies as long as the Gauss constraints are locally defined and spatially structured.

Experimental results

Research questions

• RQ1 Can local fermion-to-qubit mappings be implemented with exactly enforced constraints using only local quantum gates?
• RQ2 Can constant-depth quantum circuits be constructed for time-evolution of local fermionic Hamiltonians in d>1 dimensions?
• RQ3 How can variational quantum circuits be designed to preserve fermion number and periodicity in constrained Hilbert spaces?
• RQ4 What is the circuit depth scaling of time-evolution under local mappings compared to standard Jordan-Wigner encodings?
• RQ5 Can this framework be generalized to simulate lattice gauge theories with Gauss-law constraints?

Key findings

• The proposed quantum circuits achieve constant circuit depth per Trotter step for time-evolution, scaling as O(1) rather than O(L) as in previous approaches.
• The method exactly enforces Gauss-law constraints using local unitary operations, ensuring the physical Hilbert space is preserved.
• Ground state energies of the t-V model were successfully computed using variational quantum eigensolver (VQE) with the proposed ansatz.
• The approach enables efficient simulation of fermionic systems in 2D and higher dimensions by eliminating long-range Jordan-Wigner strings.
• The same circuit structure can be reused across different Hamiltonians on the same lattice topology, due to the topological nature of the constraints.
• The framework is extendable to other local fermion-to-qubit mappings and can be adapted to simulate lattice gauge theories with Gauss-law invariance.