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arXiv:2211.16389 · Physical review. B./Physical review. B

Ultrafast hybrid fermion-to-qubit mapping

Oliver O'Brien, Sergii Strelchuk

Published Mar 22, 2024• 12 citations
arXivHTMLPDF

Abstract

Fermion-to-qubit mappings play a crucial role in representing fermionic interactions on a quantum computer. Efficient mappings translate fermionic modes of a system to qubit interactions with a high degree of locality while using few auxiliary resources. We present a family of locality-preserving fermion-to-qubit mappings that require fewer auxiliary qubits than all existing schemes known to date. One instance requires only 1.016 qubits-per-fermion compared with 1.25 for the best-known locality-preserving mapping by Chen and Xu []. Our family of mappings (parameterized by integer n ) establishes a direct trade-off between the number of auxiliary qubits ( 1/n2 ) and the circuit length [O(logn)] . Furthermore, we present a nonlocal variant that combines the strengths of the Jordan-Wigner and Bravyi-Kitaev mappings to give 98% shorter circuits than the Jordan-Wigner mapping. This is achieved by applying seemingly incompatible mappings at different scales, making it possible for their respective strengths to complement each other. Published by the American Physical Society 2024

DOI
10.1103/physrevb.109.115149

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Fermion-to-qubit mappings play a crucial role in representing fermionic interactions on a quantum computer. Efficient mappings translate fermionic modes of a system to qubit interactions with a high degree of locality while using few auxiliary resources. We present a family of locality-preserving fermion-to-qubit mappings that require fewer auxiliary qubits than all existing schemes known to date. One instance requires only 1.016 qubits-per-fermion compared with 1.25 for the best-known locality-preserving mapping by Chen and Xu []. Our family of mappings (parameterized by integer n ) establishes a direct trade-off between the number of auxiliary qubits ( 1/n2 ) and the circuit length [O(logn)] . Furthermore, we present a nonlocal variant that combines the strengths of the Jordan-Wigner and Bravyi-Kitaev mappings to give 98% shorter circuits than the Jordan-Wigner mapping. This is achieved by applying seemingly incompatible mappings at different scales, making it possible for their respective strengths to complement each other. Published by the American Physical Society 2024
Simulating fermionic lattice models with qubits requires mapping fermionic degrees of freedom to qubits. The simplest method for this task, the Jordan-Wigner transformation, yields strings of Pauli operators acting on an extensive number of qubits. This overhead can be a hindrance to implementation of qubit-based quantum simulators, especially in the analog context. Here we thus review and analyze alternative fermion-to-qubit mappings, including the two approaches by Bravyi and Kitaev and the auxiliary fermion transformation. The Bravyi-Kitaev transform is reformulated in terms of a classical data structure and generalized to achieve a further locality improvement for local fermionic models on a rectangular lattice. We conclude that the most compact encoding of the fermionic operators can be done using ancilla qubits with the auxiliary fermion scheme. Without introducing ancillas, a variant of the Bravyi-Kitaev transform provides the most compact fermion-to-qubit mapping for Hubbard-like models.
We argue that all locality-preserving mappings between fermionic observables and Pauli matrices on a two-dimensional lattice can be generated from the exact bosonization in Chen et al. [Ann.Phys.(N.Y) 393, 234 (2018)], whose gauge constraints project onto the subspace of the toric code with emergent fermions.Starting from the exact bosonization and applying Clifford finite-depth generalized local unitary transformation, we can achieve all possible fermion-to-qubit mappings (up to the re-pairing of Majorana fermions).In particular, we discover a new supercompact encoding using 1.25 qubits per fermion on the square lattice.We prove the existence of finite-depth quantum circuits to obtain fermion-to-qubit mappings with qubit-fermion ratios r = 1 + 1/2k for positive integers k, utilizing the trivialness of quantum cellular automata in two spatial dimensions.Also, we provide direct constructions of fermion-to-qubit mappings with ratios arbitrarily close to 1.When the ratio reaches 1, the fermion-to-qubit mapping reduces to the one-dimensional Jordan-Wigner transformation along a certain path in the two-dimensional lattice.Finally, we explicitly demonstrate that the Bravyi-Kitaev superfast simulation, the Verstraete-Cirac auxiliary method, Kitaev's exactly solved model, the Majorana loop stabilizer codes, and the compact fermion-to-qubit mapping can all be obtained from the exact bosonization.
Quantum simulation of fermionic systems is a promising application of quantum computers, but to program them, we need to map fermionic states and operators to qubit states and quantum gates. While quantum processors may be built as two-dimensional qubit networks with couplings between nearest neighbors, standard fermion-to-qubit mappings do not account for that kind of connectivity. In this work we concatenate the (one-dimensional) Jordan-Wigner transform with specific quantum codes defined under the addition of a certain number of auxiliary qubits. This yields a class of mappings with which any fermionic system can be embedded in a two-dimensional qubit setup, fostering scalable quantum simulation. Our technique is demonstrated on the two-dimensional Fermi-Hubbard model, which we transform into a local Hamiltonian. What is more, we adapt the Verstraete-Cirac transform and Bravyi-Kitaev superfast simulation to the square lattice connectivity and compare them to our mappings. An advantage of our approach in this comparison is that it allows us to encode and decode a logical state with a simple unitary quantum circuit.
Digital quantum simulation of fermionic systems is important in the context of chemistry and physics. Simulating fermionic models on general purpose quantum computers requires imposing a fermionic algebra on qubits. The previously studied Jordan-Wigner and Bravyi-Kitaev transformations are two techniques for accomplishing this task. Here, we reexamine an auxiliary fermion construction which maps fermionic operators to local operators on qubits. The local simulation is performed by relaxing the requirement that the number of qubits should match the number of single-particle states. Instead, auxiliary sites are introduced to enable nonconsecutive fermionic couplings to be simulated with constant low-rank tensor products on qubits. The additional number of auxiliary qubits required per fermionic degree of freedom depends only on the degree of connectivity of the Hamiltonian. We connect the auxiliary fermion construction to topological models and give examples of the construction.
Simulation of fermionic systems is one of the most promising applications of quantum computers. It spans problems in quantum chemistry, high-energy physics and condensed matter. Underpinning the core steps of any quantum simulation algorithm, fermion-qubit mappings translate the fermionic interactions to the operators and states of quantum computers. This translation is highly non-trivial: a burgeoning supply of fermion-qubit mappings has arisen over the past twenty years to address the limited resources of early quantum technology. Previous literature has presented a dichotomy between ancillafree fermion-qubit mappings, which minimise qubit count, and local encodings, which minimise gate complexity. We present two computational approaches to the construction of general mappings while working with a limited number of qubits, striking a balance between the low-qubit and low-gate demands of present quantum technology. The first method frames the order of fermionic labels as an instance of the quadratic assignment problem to minimize the total and maximum Pauli weights in a problem Hamiltonian. We compare the orderoptimized performance of several common ancilla-free mappings on systems of size up to 225 fermionic modes. The second method is a computational approach to incrementally add ancilla qubits to Jordan-Wigner transformations and further reduce the Pauli weights. By adding up to 10 ancilla qubits, we were able to reduce the total Pauli weight by as much as 67 % in JordanWigner transformations of fermionic systems with up to 64 modes, outperforming the previous state-of-the-art ancilla-free mappings. Reproducibility: source code and data are available at https://github.com/cameton/QCE_QubitAssignment
We investigate the utility of the locality-preserving Derby-Klassen (DK) fermion-to-qubit mapping [arXiv:2003.06939] for variational quantum simulation of two-dimensional ttt-VVV and Fermi-Hubbard models. The DK mapping preserves the locality of fermionic interactions with an enlarged Hilbert space, thereby requiring additional constraints that define the physical sector. We incorporate these constraints directly into a Hamiltonian Variational Ansätz (HVA) through Clifford-gate state preparation and use the Variational Quantum Eigensolver (VQE) to show that the low-energy properties of the resulting qubit Hamiltonian are accurately reproduced. We further exploit particle-number conservation inherent in the ansätz to resolve distinct symmetry sectors and reliably access degenerate states. Consequently, we benchmark the DK-HVA against Jordan-Wigner-based variational circuits at nonzero chemical potential, where particle-hole symmetry and the associated half-filled sign-free condition are absent. Finally, we demonstrate the advantage of locality-preserving mappings in higher-dimensional fermionic systems, where the conventional Jordan-Wigner (JW) transformation generates increasingly long Pauli strings and corresponding circuit overheads. We further extend the framework to the spinful Fermi-Hubbard model and identify a tradeoff between fermionic-mode placement and the locality of hopping and on-site interaction terms. These results establish a practical framework combining locality-preserving fermion-to-qubit mappings, constraint-preserving Clifford state preparation, and symmetry-preserving variational circuits, that trades auxiliary-qubit and stabilizer-preparation overhead for reduced operator nonlocality, providing a practical route toward resource-efficient quantum simulation of higher-dimensional interacting fermionic systems.