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Progress in High Energy Physics PHEP

Qubitization of Bosons

Desheng Li, Xinyu Huang, Lang Yu, Xu Lu, Yin Yang, Chunwang Wu, Wei Wu, Pingxing Chen

Published May 20, 2025• 0 citations
Publisher

Abstract

A binary mapping from Fock space of bosonic state to qubits is given. Based on the binary mapping, analgorithm of qubitization of bosons with complexity O(log(N)) is described by recurrence relation. As anproposal, the algorithm of qubitization of bosons to simulate real time dynamics of Yukawa coupling isrealized which might be achieved in superconductivity noisy intermediate–scale quantum computer atnot far future.

DOI
10.31526/phep.2025.03

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A binary mapping from Fock space of bosonic state to qubits is given. Based on the binary mapping, analgorithm of qubitization of bosons with complexity O(log(N)) is described by recurrence relation. As anproposal, the algorithm of qubitization of bosons to simulate real time dynamics of Yukawa coupling isrealized which might be achieved in superconductivity noisy intermediate–scale quantum computer atnot far future.
We show that one can quantum simulate the dynamics of a Sachdev-Ye-Kitaev model with NNN Majorana modes for time ttt to precision $\ensuremath{\epsilon}$ with gate complexity $O({N}^{7/2}t+{N}^{5/2}t\phantom{\rule{0.16em}{0ex}}\mathrm{polylog}(N/\ensuremath{\epsilon}))$. In addition to scaling sublinearly in the number of Hamiltonian terms, this gate complexity represents an exponential improvement in $1/\ensuremath{\epsilon}$ and large polynomial improvement in NNN and ttt over prior state-of-the-art algorithms which scale as $O({N}^{10}{t}^{2}/\ensuremath{\epsilon})$. Our approach involves a variant of the qubitization technique in which we encode the Hamiltonian HHH as an asymmetric projection of a signal oracle UUU onto two different signal states prepared by state oracles, $A|0\ensuremath{\rangle}\ensuremath{\mapsto}|A\ensuremath{\rangle}$ and $B|0\ensuremath{\rangle}\ensuremath{\mapsto}|B\ensuremath{\rangle}$, such that $H=\ensuremath{\langle}B|U|A\ensuremath{\rangle}$. Our strategy for applying this method to the Sachdev-Ye-Kitaev model involves realizing BBB using only Hadamard gates and realizing AAA as a random quantum circuit.
Superconducting circuits based on Josephson junctions are regarded as one of the most promising technologies for the implementation of scalable quantum computers. This review presents the basic principles of superconducting qubits and shows the progress of quantum computing and quantum simulation based on superconducting qubits in recent years. The experimental realization of gate operations, readout, error correction codes, as well as some quantum algorithms are summarized, followed by an introduction of quantum simulation. And then some important applications in fields including condensed matter physics, quantum annealing, and quantum chemistry are discussed.
Classical simulations of real-space quantum dynamics are challenging due to the exponential scaling of computational cost with system dimensions. Quantum computers offer the potential to simulate quantum dynamics with polynomial complexity; however, existing quantum algorithms based on the split-operator techniques require large-scale fault-tolerant quantum computers that remain elusive in the near future. Here, we present variational simulations of real-space quantum dynamics suitable for implementation in noisy intermediate-scale quantum (NISQ) devices. The Hamiltonian is first encoded onto qubits using a discrete variable representation and binary encoding scheme. We show that direct application of a real-time variational quantum algorithm based on the McLachlan's principle is inefficient as the measurement cost grows exponentially with the qubit number for general potential energy, and an extremely small time-step size is required to achieve accurate results. Motivated by the insights that many chemical dynamics occur in the low-energy subspace, we propose a subspace expansion method by projecting the total Hamiltonian, including the time-dependent driving field, onto the system low-energy eigenstate subspace using quantum computers, and the exact quantum dynamics within the subspace can then be solved classically. We show that the measurement cost of the subspace approach grows polynomially with dimensionality for general potential energy. Our numerical examples demonstrate the capability of our approach, even under intense laser fields. Our work opens the possibility of simulating chemical dynamics with NISQ hardware.
Abstract The Lindblad equation generalizes the Schrödinger equation to quantum systems that undergo dissipative dynamics. The quantum simulation of Lindbladian dynamics is therefore non-unitary, preventing a naive application of state-of-the-art quantum algorithms. Here, we make use of an approximate correspondence between Lindbladian dynamics and evolution based on repeated interaction (RI) CPTP maps to write down a Hamiltonian formulation of the Lindblad dynamics and derive a rigorous error bound on the master equation. Specifically, we show that the number of interactions needed to simulate the Liouvillian e t L within error ε scales in most physical scenarios as ν ∈ O ( t 2 ‖ L ‖ 1 → 1 2 / ϵ ) . This is significant because the error in the Lindbladian approximation to the dynamics is not explicitly bounded in existing quantum algorithms for open system simulations. We then provide quantum algorithms to simulate RI maps using an iterative qubitization approach and Trotter–Suzuki formulas, and specifically show that for iterative qubitization the number of operations needed to simulate the dynamics (for a fixed value of ν) scales as O ( α 0 t + ν log ⁡ ( 1 / ϵ ) / log ⁡ log ⁡ ( 1 / ϵ ) ) in the limit where α 0 (the coefficient 1-norm for the system and bath Hamiltonians) asymptotically dominates over the corresponding factor for the interaction Hamiltonian, which is often the case in weak coupling. This scaling would appear to be optimal if the complexity of ν is not considered, which underscores the importance of considering the error in the Liouvillian that we reveal in this work.