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arXiv:2105.10941 · Physical Review A

Quantum simulation of second-quantized Hamiltonians in compact encoding

William M. Kirby, Sultana Hadi, Michael Kreshchuk, Peter J. Love

Published Oct 13, 2021• 19 citations
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Abstract

We describe methods for simulating general second-quantized Hamiltonians using the compact encoding, in which qubit states encode only the occupied modes in physical occupation number basis states. These methods apply to second-quantized Hamiltonians composed of a constant number of interactions, i.e., linear combinations of ladder operator monomials of fixed form. Compact encoding leads to qubit requirements that are optimal up to logarithmic factors. We show how to use sparse Hamiltonian simulation methods for second-quantized Hamiltonians in compact encoding, give explicit implementations for the required oracles, and analyze the methods. We also describe several example applications including the free boson and fermion theories, the ${\ensuremath{\phi}}^{4}$-theory, and the massive Yukawa model, all in both equal-time and light-front quantization. Our methods provide a general-purpose tool for simulating second-quantized Hamiltonians, with optimal or near-optimal scaling with error and model parameters.

DOI
10.1103/physreva.104.042607

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We describe methods for simulating general second-quantized Hamiltonians using the compact encoding, in which qubit states encode only the occupied modes in physical occupation number basis states. These methods apply to second-quantized Hamiltonians composed of a constant number of interactions, i.e., linear combinations of ladder operator monomials of fixed form. Compact encoding leads to qubit requirements that are optimal up to logarithmic factors. We show how to use sparse Hamiltonian simulation methods for second-quantized Hamiltonians in compact encoding, give explicit implementations for the required oracles, and analyze the methods. We also describe several example applications including the free boson and fermion theories, the \ensuremathϕ4{\ensuremath{\phi}}^{4}\ensuremathϕ4-theory, and the massive Yukawa model, all in both equal-time and light-front quantization. Our methods provide a general-purpose tool for simulating second-quantized Hamiltonians, with optimal or near-optimal scaling with error and model parameters.
We propose a framework for computing the structure and dynamics for second-quantized many-nucleon Hamiltonians on quantum computers. We develop an oracle-based Hamiltonian input model that computes the many-nucleon states and nonzero Hamiltonian matrix elements of the many-nucleon system. With our Fock-state based input model, we show how to implement the sparse matrix simulation algorithms to calculate the dynamics of the second-quantized many-nucleon Hamiltonian. Based on the dynamics simulation methods, we also present the methodology for structure calculations of the many-nucleon system. In this work, we provide an explicit circuit design of our input model of the second-quantized Hamiltonian within a direct encoding scheme that maps the occupation of each available single-particle state in the many-nucleon state to the state of specific qubit in a quantum register. We analyze our method and provide the asymptotic cost in computing resources for structure and dynamics calculations of many-nucleon systems. For pedagogical purposes, we demonstrate our input model with two model problems in restricted model spaces.
We present a method for encoding second-quantized fermionic systems in qubits when the number of fermions is conserved, as in the electronic structure problem. When the number FFF of fermions is much smaller than the number MMM of modes, this symmetry reduces the number of information-theoretically required qubits from Θ(M)\Theta(M)Θ(M) to O(Flog⁡M)O(F\log M)O(FlogM). In this limit, our encoding requires O(F2log⁡4M)O(F^2\log^4 M)O(F2log4M) qubits, while encoded fermionic creation and annihilation operators have cost O(F2log⁡5M)O(F^2\log^5 M)O(F2log5M) in two-qubit gates. This is the first second-quantized encoding of fermions in qubits whose costs in qubits and gates are both polylogarithmic in MMM, which will permit studying fermionic systems in the high-precision regime of many modes.
Efficient block encoding of many-body Hamiltonians is a central requirement for quantum algorithms in scientific computing, particularly in the early fault-tolerant era. In this work, we introduce new explicit constructions for block encoding second-quantized Hamiltonians that substantially reduce Clifford+T gate complexity and ancilla overhead. By utilizing a data lookup strategy based on the SWAP architecture for the sparsity oracle OCO_COC​, and a direct sampling method for the amplitude oracle OAO_AOA​ with SELECT-SWAP architecture, we achieve a T count that scales as O~(L)\mathcal{\tilde{O}}(\sqrt{L})O~(L​) with respect to the number of interaction terms LLL in general second-quantized Hamiltonians. We also achieve an improved constant factor in the Clifford gate count of our oracle. Furthermore, we design a block encoding that directly targets the ηηη-particle subspace, thereby reducing the subnormalization factor from O(L)\mathcal{O}(L)O(L) to O(L)\mathcal{O}(\sqrt{L})O(L​), and improving fault-tolerant efficiency when simulating systems with fixed particle numbers. Building on the block encoding framework developed for general many-body Hamiltonians, we extend our approach to electronic Hamiltonians whose coefficient tensors exhibit translation invariance or possess decaying structures. Our results provide a practical path toward early fault-tolerant quantum simulation of many-body systems, substantially lowering resource overheads compared to previous methods.
Efficient simulation of interacting fermionic systems is a key application of near-term quantum computers, but is hindered by the overhead required to encode fermionic operators on qubit hardware. Here, we consider models with NNN fermionic modes in which each participates in at most a constant number ddd of interactions and study the circuit depth required to implement the Trotterized time evolution on qubit hardware with all-to-all connectivity. We introduce an encoding that augments each physical fermionic mode with a small number of auxiliary fermions, enabling the removal of Jordan--Wigner strings. Although the preparation of the auxiliary fermion state incurs an initial overhead, this state remains invariant under time evolution. As a result, long-time evolution can be implemented with asymptotically optimal circuit depth, reducing a previously multiplicative O(log⁡N)O(\log N)O(logN) overhead to an additive overhead. Our results thus establish that the simulation of sparse fermionic models on qubit hardware matches the performance achievable on ideal fermionic hardware up to constant factors and O(dN)O(dN)O(dN) ancillary qubits.
Simulating electronic structure on a quantum computer requires encoding of fermionic systems onto qubits. Common encoding methods transform a fermionic system of NNN spin-orbitals into an NNN-qubit system, but many of the fermionic configurations do not respect the required conditions and symmetries of the system so the qubit Hilbert space in this case may have unphysical states and thus cannot be fully utilized. We propose a generalized qubit-efficient encoding (QEE) scheme that requires the qubit number to be only logarithmic in the number of configurations that satisfy the required conditions and symmetries. For the case of considering only the particle-conserving and singlet configurations, we reduce the qubit count to an upper bound of O(mlog2N)\mathcal{O}(m{log}_{2}N)O(mlog2​N), where mmm is the number of particles. This QEE scheme is demonstrated on an H2{\mathrm{H}}_{2}H2​ molecule in the 6-31G basis set and a LiH\mathrm{LiH}LiH molecule in the STO-3G basis set using fewer qubits than the common encoding methods. We calculate the ground-state energy surfaces using a variational quantum eigensolver algorithm with a hardware-efficient ansatz circuit. We choose to use a hardware-efficient ansatz since most of the Hilbert space in our scheme is spanned by desired configurations so a heuristic search for an eigenstate is sensible. The simulations are performed on IBM Quantum machines and the Qiskit simulator with a noise model implemented from a IBM Quantum machine. Using the methods of measurement error mitigation and error-free linear extrapolation, we demonstrate that most of the distributions of the extrapolated energies using our QEE scheme agree with the exact results obtained by Hamiltonian diagonalization in the given basis sets within chemical accuracy. Our proposed scheme and results show the feasibility of quantum simulations for larger molecular systems in the noisy intermediate-scale quantum (NISQ) era. The number of terms in the Hamiltonian has an upper bound of $\mathcal{O}(\frac{{N}^{2m+1}}{(m\ensuremath{-}1)!\phantom{\rule{0.16em}{0ex}}m!})$ for the QEE scheme while it scales as O(N4)\mathcal{O}({N}^{4})O(N4) for the Jordan-Wigner encoding scheme. Nevertheless, we present several cases where QEE is useful.
Abstract Elementary quantum mechanics proposes that a closed physical system consistently evolves in a reversible manner. However, control and readout necessitate the coupling of the quantum system to the external environment, subjecting it to relaxation and decoherence. Consequently, system-environment interactions are indispensable for simulating physically significant theories. A broad spectrum of physical systems in condensed-matter and high-energy physics, vibrational spectroscopy, and circuit and cavity QED necessitates the incorporation of bosonic degrees of freedom, such as phonons, photons, and gluons, into optimized fermion algorithms for near-future quantum simulations. In particular, when a quantum system is surrounded by an external environment, its basic physics can usually be simplified to a spin or fermionic system interacting with bosonic modes. Nevertheless, troublesome factors such as the magnitude of the bosonic degrees of freedom typically complicate the direct quantum simulation of these interacting models, necessitating the consideration of a comprehensive plan. This strategy should specifically include a suitable fermion/boson-to-qubit mapping scheme to encode sufficiently large yet manageable bosonic modes, and a method for truncating and/or downfolding the Hamiltonian to the defined subspace for performing an approximate but highly accurate simulation, guided by rigorous error analysis. In this pedagogical tutorial review, we aim to provide such an exhaustive strategy, focusing on encoding and simulating certain bosonic-related model Hamiltonians, inclusive of their static properties and time evolutions. Specifically, we emphasize two aspects: (1) the discussion of recently developed quantum algorithms for these interacting models and the construction of effective Hamiltonians, and (2) a detailed analysis regarding a tightened error bound for truncating the bosonic modes for a class of fermion-boson interacting Hamiltonians.