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arXiv:1910.10746 · Quantum

Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning

Zhang Jiang, Amir Kalev, Wojciech Mruczkiewicz, Hartmut Neven

Published Jun 4, 2020• 81 citations
arXivHTMLPDF

Abstract

We introduce a fermion-to-qubit mapping defined on ternary trees, where any single Majorana operator on an n -mode fermionic system is mapped to a multi-qubit Pauli operator acting nontrivially on ⌈log3⁡(2n+1)⌉ qubits. The mapping has a simple structure and is optimal in the sense that it is impossible to construct Pauli operators in any fermion-to-qubit mapping acting nontrivially on less than log3⁡(2n) qubits on average. We apply it to the problem of learning k -fermion reduced density matrix (RDM), a problem relevant in various quantum simulation applications. We show that one can determine individual elements of all k -fermion RDMs in parallel, to precision ϵ , by repeating a single quantum circuit for ≲(2n+1)kϵ−2 times. This result is based on a method we develop here that allows one to determine individual elements of all k -qubit RDMs in parallel, to precision ϵ , by repeating a single quantum circuit for ≲3kϵ−2 times, independent of the system size. This improves over existing schemes for determining qubit RDMs.

DOI
10.22331/q-2020-06-04-276

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We introduce a fermion-to-qubit mapping defined on ternary trees, where any single Majorana operator on an n -mode fermionic system is mapped to a multi-qubit Pauli operator acting nontrivially on ⌈log3⁡(2n+1)⌉ qubits. The mapping has a simple structure and is optimal in the sense that it is impossible to construct Pauli operators in any fermion-to-qubit mapping acting nontrivially on less than log3⁡(2n) qubits on average. We apply it to the problem of learning k -fermion reduced density matrix (RDM), a problem relevant in various quantum simulation applications. We show that one can determine individual elements of all k -fermion RDMs in parallel, to precision ϵ , by repeating a single quantum circuit for ≲(2n+1)kϵ−2 times. This result is based on a method we develop here that allows one to determine individual elements of all k -qubit RDMs in parallel, to precision ϵ , by repeating a single quantum circuit for ≲3kϵ−2 times, independent of the system size. This improves over existing schemes for determining qubit RDMs.
This paper introduces the Hamiltonian-Adaptive Ternary Tree (HATT) framework to compile optimized Fermion-to-qubit mapping for specific Fermionic Hamiltonians. In the simulation of Fermionic quantum systems, efficient Fermion-toqubit mapping plays a critical role in transforming the Fermionic system into a qubit system. HATT utilizes ternary tree mapping and a bottom-up construction procedure to generate Hamiltonian aware Fermion-to-qubit mapping to reduce the Pauli weight of the qubit Hamiltonian, resulting in lower quantum simulation circuit overhead. Additionally, our optimizations retain the important vacuum state preservation property in our Fermion-toqubit mapping and reduce the complexity of our algorithm from O(N4)O\left(N^{4}\right)O(N4) to O(N3)O\left(N^{3}\right)O(N3). Evaluations on various Fermionic systems demonstrate 5∼25%5 \sim 25 \%5∼25% reduction in Pauli weight, gate count, and circuit depth, alongside excellent scalability to larger systems. Experiments on the Ionq device also show the advantages of HATT in noise resistance in quantum simulations.
Simulation of fermionic Hamiltonians with gate-based quantum computers requires the selection of an encoding from fermionic operators to quantum gates, the most widely used being the Jordan-Wigner transform. Many alternative encodings exist, with quantum circuits and simulation results being sensitive to choice of encoding, device connectivity and Hamiltonian characteristics. Non-stochastic optimisation of the ternary tree class of encodings to date has targeted either the device or Hamiltonian. We develop a deterministic method which optimises ternary tree encodings without changing the underlying tree structure. This enables reduction in Pauli-weight without ancillae or additional swap-gate overhead. We demonstrate this method for a variety of encodings, including those which are derived from the qubit connectivity graph of a quantum computer. Numerical results for a suite of standard encoding methods applied to water in the STO-3G basis indicate that our method reduces qDRIFT circuit depths on average by 24.7% and 26.5% for untranspiled and transpiled circuits respectively.
Simulation of interacting Fermionic Hamiltonians is one of the most promising applications of quantum computers. However, the feasibility of analyzing Fermionic systems with a quantum computer hinges on the efficiency of Fermion-to-qubit mappings that encode nonlocal Fermionic degrees of freedom in local qubit degrees of freedom. While recent studies have highlighted the importance of designing Fermion-to-qubit mappings that are tailored to specific problem Hamiltonians, the methods proposed so far either are restricted to a narrow class of mappings or they use computationally expensive and unscalable brute-force search algorithms. Here, we address this challenge by designing a heuristic numerical optimization framework for Fermion-to-qubit mappings. To this end, we first translate the Fermion-to-qubit mapping problem to a Clifford circuit optimization problem and then use simulated annealing to optimize the average Pauli weight of the problem Hamiltonian. For all Fermionic Hamiltonians we have considered, the numerically optimized mappings outperform their conventional counterparts, including ternary-tree-based mappings that are known to be optimal for single creation and annihilation operators. We find that our optimized mappings yield between 15% and 40% improvements on the average Pauli weight when the simulation Hamiltonian has an intermediate level of complexity. Most remarkably, the optimized mappings improve the average Pauli weight for 6 × 6 nearest-neighbor hopping and Hubbard models by more than 40% and 20%, respectively. Surprisingly, we also find specific interaction Hamiltonians for which the optimized mapping outperforms any ternary-tree-based mapping. Our results establish heuristic numerical optimization as an effective method for obtaining mappings tailored for specific Fermionic Hamiltonian.
Fermion-to-qubit mappings are used to represent fermionic modes on quantum computers, an essential first step in many quantum algorithms for electronic structure calculations. In this work, we present a formalism to design flexible fermion-to-qubit mappings from ternary trees. We discuss in an intuitive manner the connection between the generating trees' structure and certain properties of the resulting mapping, such as Pauli weight and the delocalisation of mode occupation. Moreover, we introduce a recipe that guarantees Fock basis states are mapped to computational basis states in qubit space, a desirable property for many applications in quantum computing. Based on this formalism, we introduce the Bonsai algorithm, which takes as input the potentially limited topology of the qubit connectivity of a quantum device and returns a tailored fermion-to-qubit mapping that reduces the SWAP overhead with respect to other paradigmatic mappings. We illustrate the algorithm by producing mappings for the heavy-hexagon topology widely used in IBM quantum computers. The resulting mappings have a favourable Pauli weight scaling O(N)\mathcal{O}(\sqrt{N})O(N​) on this connectivity, while ensuring that no SWAP gates are necessary for single excitation operations.
Fermion-to-qubit mappings are used to represent fermionic modes on quantum computers, an essential first step in many quantum algorithms for electronic structure calculations.In this work, we present a formalism to design flexible fermion-to-qubit mappings from ternary trees.We intuitively discuss the connection between the structure of the generating trees and certain properties of the resulting mapping, such as the Pauli weight and the delocalization of mode occupation.Moreover, we introduce a recipe that guarantees that Fock basis states are mapped to computational basis states in qubit space, a desirable property for many applications in quantum computing.Based on this formalism, we introduce the Bonsai algorithm, which takes as input the potentially limited topology of the qubit connectivity of a quantum device and returns a tailored fermion-to-qubit mapping that reduces the SWAP overhead compared to other paradigmatic mappings.We illustrate the algorithm by producing mappings for the heavy-hexagon topology widely used in IBM quantum computers.The resulting mappings have a favorable Pauli-weight scaling O( √ N ) on this connectivity while ensuring that no SWAP gates are necessary for single-excitation operations.
Mapping fermionic systems to qubits on a quantum computer is often the first step for algorithms in quantum chemistry and condensed matter physics. However, it is difficult to reconcile the many different approaches that have been proposed, such as those based on binary matrices, ternary trees, and stabilizer codes. This challenge is further exacerbated by the many ways to describe them -- transformation of Majorana operators, action on Fock states, encoder circuits, and stabilizers of local encodings -- making it challenging to know when the mappings are equivalent. In this work, we present a graphical framework for fermion-to-qubit mappings that streamlines and unifies various representations through the ZX-calculus. To start, we present the correspondence between linear encodings of the Fock basis and phase-free ZX-diagrams. The commutation rules of scalable ZX-calculus allows us to convert the fermionic operators to Pauli operators under any linear encoding. Next, we give a translation from ternary tree mappings to scalable ZX-diagrams, which not only directly represents the encoder map as a CNOT circuit, but also retains the same structure as the tree. Consequently, we graphically prove that ternary tree transformations are equivalent to linear encodings, a recent result by Chiew et al. The scalable ZX representation moreover enables us to construct an algorithm to directly compute the binary matrix for any ternary tree mapping. Lastly, we present the graphical representation of local fermion-to-qubit encodings. Its encoder ZX-diagram has the same connectivity as the interaction graph of the fermionic Hamiltonian and also allows us to easily identify stabilizers of the encoding.