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arXiv:2404.05810

Ground State Preparation via Dynamical Cooling

Danial Motlagh, Modjtaba Shokrian Zini, Juan Miguel Arrazola, Nathan Wiebe

Published Apr 8, 2024• 1 citation
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Abstract

Quantum algorithms for probing ground-state properties of quantum systems require good initial states. Projection-based methods such as eigenvalue filtering rely on inputs that have a significant overlap with the low-energy subspace, which can be challenging for large, strongly-correlated systems. This issue has motivated the study of physically-inspired dynamical approaches such as thermodynamic cooling. In this work, we introduce a ground-state preparation algorithm based on the simulation of quantum dynamics. Our main insight is to transform the Hamiltonian by a shifted sign function via quantum signal processing, effectively mapping eigenvalues into positive and negative subspaces separated by a large gap. This automatically ensures that all states within each subspace conserve energy with respect to the transformed Hamiltonian. Subsequent time-evolution with a perturbed Hamiltonian induces transitions to lower-energy states while preventing unwanted jumps to higher energy states. The approach does not rely on a priori knowledge of energy gaps and requires no additional qubits to model a bath. Furthermore, it makes $\tilde{\mathcal{O}}(d^{\,3/2}/ε)$ queries to the time-evolution operator of the system and $\tilde{\mathcal{O}}(d^{\,3/2})$ queries to a block-encoding of the perturbation, for $d$ cooling steps and an $ε$-accurate energy resolution. Our results provide a framework for combining quantum signal processing and Hamiltonian simulation to design heuristic quantum algorithms for ground-state preparation.

DOI
10.48550/arxiv.2404.05810

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Quantum algorithms for probing ground-state properties of quantum systems require good initial states. Projection-based methods such as eigenvalue filtering rely on inputs that have a significant overlap with the low-energy subspace, which can be challenging for large, strongly-correlated systems. This issue has motivated the study of physically-inspired dynamical approaches such as thermodynamic cooling. In this work, we introduce a ground-state preparation algorithm based on the simulation of quantum dynamics. Our main insight is to transform the Hamiltonian by a shifted sign function via quantum signal processing, effectively mapping eigenvalues into positive and negative subspaces separated by a large gap. This automatically ensures that all states within each subspace conserve energy with respect to the transformed Hamiltonian. Subsequent time-evolution with a perturbed Hamiltonian induces transitions to lower-energy states while preventing unwanted jumps to higher energy states. The approach does not rely on a priori knowledge of energy gaps and requires no additional qubits to model a bath. Furthermore, it makes O~(d 3/2/ε)\tilde{\mathcal{O}}(d^{\,3/2}/ε)O~(d3/2/ε) queries to the time-evolution operator of the system and O~(d 3/2)\tilde{\mathcal{O}}(d^{\,3/2})O~(d3/2) queries to a block-encoding of the perturbation, for ddd cooling steps and an εεε-accurate energy resolution. Our results provide a framework for combining quantum signal processing and Hamiltonian simulation to design heuristic quantum algorithms for ground-state preparation.
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The preparation of the ground state of a Hamiltonian HHH with a large spectral radius has applications in many areas such as electronic structure theory and quantum field theory. Given an initial state with a constant overlap with the ground state, and assuming that the Hamiltonian HHH can be efficiently simulated with an ideal fast-forwarding protocol, we first demonstrate that employing a linear combination of unitaries (LCU) approach can prepare the ground state at a cost of O(log⁡2(∥H∥Δ−1))\mathcal{O}(\log^2(\|H\| Δ^{-1}))O(log2(∥H∥Δ−1)) queries to controlled Hamiltonian evolution. Here ∥H∥\|H\|∥H∥ is the spectral radius of HHH and ΔΔΔ the spectral gap. However, traditional Quantum Signal Processing (QSP)-based methods fail to capitalize on this efficient protocol, and its cost scales as O(∥H∥Δ−1)\mathcal{O}(\|H\| Δ^{-1})O(∥H∥Δ−1). To bridge this gap, we develop a multi-level QSP-based algorithm that exploits the fast-forwarding feature. This novel algorithm not only matches the efficiency of the LCU approach when an ideal fast-forwarding protocol is available, but also exceeds it with a reduced cost that scales as O(log⁡(∥H∥Δ−1))\mathcal{O}(\log(\|H\| Δ^{-1}))O(log(∥H∥Δ−1)). Additionally, our multi-level QSP method requires only O(log⁡(∥H∥Δ−1))\mathcal{O}(\log(\|H\| Δ^{-1}))O(log(∥H∥Δ−1)) coefficients for implementing single qubit rotations. This eliminates the need for constructing the PREPARE oracle in LCU, which prepares a state encoding O(∥H∥Δ−1)\mathcal{O}(\|H\| Δ^{-1})O(∥H∥Δ−1) coefficients regardless of whether the Hamiltonian can be fast-forwarded.
We consider the task of simulating time evolution under a Hamiltonian H within its low-energy subspace. Assuming access to a block-encoding of H ′ := ( H − E ) / λ , for some λ > 0 and E ∈ R , the goal is to implement an ϵ -approximation to the evolution operator e − i t H when the initial state is confined to the subspace corresponding to eigenvalues [ − 1 , − 1 + Δ / λ ] of H ′ , for Δ ≤ λ . We present a quantum algorithm that requires O ( t λ Γ + λ / Γ log ⁡ ( 1 / ϵ ) ) queries to the block-encoding for any choice of Γ such that Δ ≤ Γ ≤ λ . When the parameters satisfy log ⁡ ( 1 / ϵ ) = o ( t λ ) and Δ / λ = o ( 1 ) , this result improves over generic methods with query complexity Ω ( t λ ) . Our quantum algorithm leverages spectral gap amplification and the quantum singular value transform.For a given H , the block-encoding of its H ′ must be prepared efficiently to achieve an asymptotic speedup in simulating the low-energy subspace; we refer to these Hamiltonians as g a p − a m p l i f i a b l e . We show necessary and sufficient conditions for gap amplifiability in terms of an operationally useful decomposition of H
We introduce a method for digital preparation of ground states of simulated Hamiltonians, inspired by cooling in nature and adapted to leverage the capabilities of digital quantum hardware. The cold bath is simulated by a single ancillary qubit, which is reset periodically and coupled to the system nonperturbatively. Studying this cooling method on a 1-qubit system toy model, we optimize two cooling protocols based on weak-coupling and strong-coupling approaches. Extending the insight from the 1-qubit system model, we develop two scalable protocols for larger systems. The LogSweep protocol extends the weak-coupling approach by sweeping energies to resonantly match any targeted transition. We test LogSweep on the 1D transverse-field Ising model, demonstrating approximate ground-state preparation with an error that can be made polynomially small in the computation time for all three phases of the system. The BangBang protocol extends the strong-coupling approach, and exploits a heuristics for local Hamiltonians to maximize the probability of deexciting system transitions in the shortest possible time. Although this protocol does not promise long-time convergence, it allows for a rapid cooling to an approximation of the ground state, making this protocol appealing for near-term demonstrations.
We propose a dynamical approach to identifying ground-state quantum phases through short-time dissipative cooling. Rather than determining the phase by preparing highly accurate approximations to ground states, we prepare a representative state of a candidate phase and monitor the early-time response of phase-sensitive observables under cooling dynamics tailored to the target Hamiltonian. For a class of phase-decision problems in which the relevant observables can be inferred from the low-energy manifold, and with jump operators implementable using only short-time Hamiltonian simulation, the dissipative evolution rapidly suppresses high-energy components and drives the system into a low-energy manifold whose observables already reveal the underlying ground-state phase, well before mixing to the steady state. We demonstrate this strategy for the frustrated J1J_1J1​--J2J_2J2​ Heisenberg chain, the Kitaev honeycomb model, and the XXZ chain, including Berezinskii--Kosterlitz--Thouless and topological phase transitions. In particular, coarse filter resolutions and short evolution times suffice to recover phase-sensitive quantities such as the Luttinger parameter and topological diagnostics. We further provide theoretical justification that cooling dynamics with such jump operators can rigorously prepare low-energy manifolds for free-fermionic and free-bosonic systems, and investigate this mechanism for interacting fermionic systems. Our results suggest that phase decision is a plausible target for future utility-scale studies on early fault-tolerant quantum devices.
We present a cooling algorithm for ground state preparation of fermionic Hamiltonians. Our algorithm makes use of the Hamiltonian simulation of the considered system coupled to an ancillary fridge, which is regularly reset to its known ground state. We derive suitable interaction Hamiltonians that originate from ladder operators of the free theory and initiate resonant gaps between system and fridge. We further propose a spectroscopic scan to find the relevant eigenenergies of the system using energy measurements on the fridge. With these insights, we design a ground state cooling algorithm for fermionic systems that is efficient, i.e. its runtime is polynomial in the system size, as long as the initial state is prepared in a low-energy sector of polynomial size. We achieve the latter via a pseudo-adiabatic sweep from a parameter regime whose ground state can be easily prepared. We estimate that our algorithm has a polynomial runtime for systems where the spectral gap decreases at most polynomially in system size, and is faster than the adiabatic algorithm for a large range of settings. We generalize the algorithm to prepare thermal states and demonstrate our findings on the Fermi-Hubbard model.