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Barriers to Simulation ofQuantum Chemistry onQuantum Hardware.

Benjamin C. A Morrison

Published Dec 13, 2021• 0 citations
OpenAlex

Abstract

Inconvenient Implementation Details• The systems we're trying to simulate are made of electrons, which are indistinguishable fermions.• (Almost) any quantum computer we're likely to use to simulate them is going to be made of qubits, which are distinguishable two-level systems.• To map one onto another, we will need to correct the shape of our Hilbert space.• This has a cost in overhead, either increased weight/nonlocality of operators (Jordan-Wigner, Bravyi-Kitaev, Parity Basis) or extra qubits (BK Superfast).• In the former case, we multiply our circuit depth/time cost by O(n) or O(log n) depending on the connectivity of our hardware, where n is the number of orbitals considered.• In the latter case, we multiply our qubit count/space cost by the degree of the interaction graph, which in the general case is O(n).• If we had a quantum computer made of fermions instead of qubits, this would be free… 5

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Inconvenient Implementation Details• The systems we're trying to simulate are made of electrons, which are indistinguishable fermions.• (Almost) any quantum computer we're likely to use to simulate them is going to be made of qubits, which are distinguishable two-level systems.• To map one onto another, we will need to correct the shape of our Hilbert space.• This has a cost in overhead, either increased weight/nonlocality of operators (Jordan-Wigner, Bravyi-Kitaev, Parity Basis) or extra qubits (BK Superfast).• In the former case, we multiply our circuit depth/time cost by O(n) or O(log n) depending on the connectivity of our hardware, where n is the number of orbitals considered.• In the latter case, we multiply our qubit count/space cost by the degree of the interaction graph, which in the general case is O(n).• If we had a quantum computer made of fermions instead of qubits, this would be free… 5
Present quantum computers often work with distinguishable qubits as their computational units. In order to simulate indistinguishable fermionic particles, it is first required to map the fermionic state to the state of the qubits. The Bravyi-Kitaev Superfast (BKSF) algorithm can be used to accomplish this mapping. The BKSF mapping has connections to quantum error correction and opens the door to new ways of understanding fermionic simulation in a topological context. Here, we present the first detailed exposition of the BKSF algorithm for molecular simulation. We provide the BKSF transformed qubit operators and report on our implementation of the BKSF fermion-to-qubits transform in OpenFermion. In this initial study of a hydrogen molecule we have compared BKSF, Jordan-Wigner, and Bravyi-Kitaev transforms under the Trotter approximation. The gate count to implement BKSF is lower than Jordan-Wigner but higher than Bravyi-Kitaev. We considered different orderings of the exponentiated terms and found lower Trotter errors than the previously reported for Jordan-Wigner and Bravyi-Kitaev algorithms. These results open the door to the further study of the BKSF algorithm for quantum simulation.
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.
Simulating molecular systems on quantum computers requires efficient mappings from Fermionic operators to qubit operators. Traditional mappings such as Jordan-Wigner or Bravyi-Kitaev often produce high-weight Pauli terms, increasing circuit depth and measurement complexity. Although several local qubit mappings have been proposed to address this challenge, most are specialized for structured models like the Hubbard Hamiltonian and perform poorly for realistic chemical systems with dense two-body interactions. In this work, we utilize a suite of techniques to construct compact and noise-resilient Fermion-to-qubit mappings suitable for general molecular Hamiltonians. Building on the Generalized Superfast Encoding (GSE) and other similar works, we demonstrate that it outperforms prior encodings in both accuracy and hardware efficiency for molecular simulations. Our improvements include path optimization within the Hamiltonian's interaction graph to minimize operator weight, introduction of multi-edge graph structures for enhanced error detection without added circuit depth, and a stabilizer measurement framework that directly maps logical terms and stabilizers to the Z-basis using Clifford simulation. Applying these methods to simulations of (H2)2(H_2)_2(H2​)2​ and (H2)3(H_2)_3(H2​)3​ systems yields significantly improved absolute and correlation energy estimates under realistic hardware noise, with further accuracy gains achieved by increasing code distance. We also propose a [[2N, N, 2]] variant of GSE compatible with square-lattice and (quasi-)linear hardware topologies, demonstrating a twofold reduction in RMSE for orbital rotations on IBM Kingston hardware. These results establish GSE as a very attractive mapping for molecular quantum simulations.
Quantum computers are expected to become a powerful tool for studying physical quantum systems. Consequently, a number of quantum algorithms for studying the physical properties of such systems have been developed. While qubit-based quantum computers are naturally suited to the study of spin-1/2 systems, systems containing other degrees of freedom must first be encoded into qubits. Transformations to and from fermionic degrees of freedom have long been an important tool in physics and, now the simulation of fermionic systems on quantum computers based on qubits provides yet another application. In this perspective, we review methods for encoding fermionic degrees of freedom into qubits and attempt to dispel the persistent notion that fermionic systems beyond one dimension are fundamentally more difficult to deal with.