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arXiv:1810.05274 · Physical Review Research

Superfast encodings for fermionic quantum simulation

Kanav Setia, Sergey B. Bravyi, Antonio Mezzacapo, James Daniel Whitfield

Published Oct 18, 2019• 109 citations
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Abstract

Fermionic quantum simulation often require transforming the fermionic Hamiltonian to qubit operators. The authors provide a fermion-to-qubit encoding that simplifies the qubit Hamiltonian by introducing redundant degrees of freedom. This can be used to correct all single qubit errors in the simulation.
DOI
10.1103/physrevresearch.1.033033

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Abstract Efficient encoding of electronic operators into qubits is essential for quantum chemistry simulations. Most of the methods treat Fermionic degrees of freedom and qubits in a one-to-one fashion, handling their interactions. Alternatively, pairs of electrons can be represented as quasi-particles and encoded into qubits, significantly simplifying calculations. This work presents a Hybrid Encoding that allows splitting the Fock space into Fermionic and Bosonic subspaces. By leveraging the strengths of both approaches, we provide a flexible framework for optimizing quantum simulations based on molecular characteristics and hardware constraints.
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We discuss encodings of fermionic many-body systems by qubits in the presence of symmetries. Such encodings eliminate redundant degrees of freedom in a way that preserves a simple structure of the system Hamiltonian enabling quantum simulations with fewer qubits. First we consider U(1)U(1)U(1) symmetry describing the particle number conservation. Using a previously known encoding based on the first quantization method a system of MMM fermi modes with NNN particles can be simulated on a quantum computer with Q=Nlog⁡(M)Q=N\log{(M)}Q=Nlog(M) qubits. We propose a new version of this encoding tailored to variational quantum algorithms. Also we show how to improve sparsity of the simulator Hamiltonian using orthogonal arrays. Next we consider encodings based on the second quantization method. It is shown that encodings with a given filling fraction ν=N/Mν=N/Mν=N/M and a qubit-per-mode ratio η=Q/M<1η=Q/M<1η=Q/M<1 can be constructed from efficiently decodable classical LDPC codes with the relative distance 2ν2ν2ν and the encoding rate 1−η1-η1−η. A family of codes based on high-girth bipartite graphs is discussed. Graph-based encodings eliminate roughly M/NM/NM/N qubits. Finally we consider discrete symmetries, and show how to eliminate qubits using previously known encodings, illustrating the technique for simple molecular-type Hamiltonians.
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.
In our recent work, we examined various fermion-to-qubit mappings in the context of quantum simulation including the original Bravyi-Kitaev superfast encoding (OSE) as well as a generalized version (GSE). We return to OSE and compare it against the Jordan-Wigner (JW) transform for quantum chemistry considering the number of qubits required, the Pauli weight of terms in the transformed Hamiltonians, and the L1{L}_{1}L1​ norm of the Hamiltonian. We considered a test set of molecular systems known as the Atomization Energy 6 (AE6) as well as hydrogen lattices. Our results showed that the resource efficiency of OSE is strongly affected by the spatial locality of the underlying single-particle basis. We find that OSE is outperformed by JW when the orbitals in the underlying single-particle basis are highly overlapping, which limits its applicability to near-term quantum chemistry simulations utilizing standard basis sets. In contrast, when orbitals are overlapping with only few others, as is the case of hydrogen lattices with very tight orbitals, OSE fares comparatively better. Our results illustrate the importance of choosing the right combination of basis sets and fermion-to-qubit mapping to get the most out of a quantum device when simulating physical systems.