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arXiv:2503.16663

Gadgets for simulating a non-native $XX$ interaction in quantum annealing

Banks, Robert J., Natasha Feinstein, Roopayan Ghosh, Sougato Bose, Paul A. Warburton

Published Mar 20, 2025• 0 citations
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Abstract

In certain scenarios, quantum annealing can be made more efficient by additional $XX$ interactions. It has been shown that the additional interactions can reduce the scaling of perturbative crossings. In traditional annealing devices these couplings do not exist natively. In this work, we develop two gadgets to achieve this: a three-body gadget that requires a strong $ZZZ$ interaction; and a one-hot gadget that uses only local $X$ drives and two-body $ZZ$ interactions. The gadgets partition the Hilbert space to effectively generate a limited number of $XX$ interactions in the low-energy subspace. We numerically verify that the one-hot gadget can mitigate a perturbative crossing on a toy problem. These gadgets establish new pathways for implementing and exploiting $XX$ interactions, enabling faster and more robust quantum annealing.

DOI
10.48550/arxiv.2503.16663

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Gadgets for simulating a non-native XXXXXX interaction in quantum annealing
In certain scenarios, quantum annealing can be made more efficient by additional XXXXXX interactions. It has been shown that the additional interactions can reduce the scaling of perturbative crossings. In traditional annealing devices these couplings do not exist natively. In this work, we develop two gadgets to achieve this: a three-body gadget that requires a strong ZZZZZZZZZ interaction; and a one-hot gadget that uses only local XXX drives and two-body ZZZZZZ interactions. The gadgets partition the Hilbert space to effectively generate a limited number of XXXXXX interactions in the low-energy subspace. We numerically verify that the one-hot gadget can mitigate a perturbative crossing on a toy problem. These gadgets establish new pathways for implementing and exploiting XXXXXX interactions, enabling faster and more robust quantum annealing.
The introduction of two-body XX interactions has been shown to improve the performance of quantum annealing in solving certain combinatorial optimisation problems and would enable the simulation of a broader class of many-body Hamiltonians. Due to experimental constraints, such interactions are typically not natively available on analogue quantum devices. To overcome this limitation we introduce the truss gadget. This is an exact construction that generates two-body XX interactions using only one-body X and Z fields and two-body ZZ interactions which are readily accessible on, for instance, superconducting quantum annealers. We experimentally validate the construction by reproducing the dynamics of a target XX Hamiltonian on a trapped-ion quantum processor. We then apply the gadget to an illustrative set of instances of the weighted maximum-independent-set problem on a superconducting quantum annealer. We empirically determine an approximate thousand-fold improvement in the probability of finding a state close to the ground state for these instances, over implementation without an XX interaction.
Perturbative gadgets are used to construct a quantum Hamiltonian whose low-energy subspace approximates a given quantum kkk-local Hamiltonian up to an absolute error ϵ\epsilonϵ. Typically, gadget constructions involve terms with large interaction strengths of order poly(ϵ−1)\text{poly}(\epsilon^{-1})poly(ϵ−1). Here we present a 2-body gadget construction and prove that it approximates a Hamiltonian of interaction strength γ=O(1)\gamma = O(1)γ=O(1) up to absolute error ϵ≪γ\epsilon\ll\gammaϵ≪γ using interactions of strength O(ϵ)O(\epsilon)O(ϵ) instead of the usual inverse polynomial in ϵ\epsilonϵ. A key component in our proof is a new condition for the convergence of the perturbation series, allowing our gadget construction to be applied in parallel on multiple many-body terms. We also discuss how to apply this gadget construction for approximating 3- and kkk-local Hamiltonians. The price we pay for using much weaker interactions is a large overhead in the number of ancillary qubits, and the number of interaction terms per particle, both of which scale as O(poly(ϵ−1))O(\text{poly}(\epsilon^{-1}))O(poly(ϵ−1)). Our strong-from-weak gadgets have their primary application in complexity theory (QMA hardness of restricted Hamiltonians, a generalized area law counterexample, gap amplification), but could also motivate practical implementations with several weak interactions simulating a much stronger quantum many-body interaction.
Abstract Fundamentally, it is believed that interactions between physical objects are two-body. Perturbative gadgets are one way to break up an effective many-body coupling into pairwise interactions: a Hamiltonian with high interaction strength introduces a low-energy space in which the effective theory appearsk-body and approximates a target Hamiltonian to within precision ϵ\epsilon ϵ ϵ . One caveat of existing constructions is that the interaction strength generally scales exponentially in the locality of the terms to be approximated. In this work we propose a many-body Hamiltonian construction which introduces only a single separate energy scale of order Θ(1/N2+δ)\Theta (1/N^{2+\delta })Θ(1/N2+δ) Θ(1/N2+δ) , for a small parameter δ>0\delta >0δ>0 δ>0 , and forNterms in the target Hamiltonian Ht=∑i=1Nhi\mathbf H_\mathrm {t}=\sum _{i=1}^N \mathbf h_iHt​=∑i=1N​hi​ Ht=∑i=1Nhi to be simulated: in its low-energy subspace, our constructed system can approximate any such target Hamiltonian Ht\mathbf H_tHt​ Ht with norm ratios r=max⁡i,j∈{1,…,N}∥hi∥/∥hj∥= O (exp⁡(exp⁡( poly N)))r=\max _{i,j\in \{1,\ldots ,N\}}\Vert \mathbf h_i\Vert / \Vert \mathbf h_j \Vert ={{\,\mathrm{O}\,}}(\exp (\exp ({{\,\mathrm{poly}\,}}N)))r=maxi,j∈{1,…,N}​∥hi​∥/∥hj​∥=O(exp(exp(polyN))) r=maxi,j∈{1,…,N}‖hi‖/‖hj‖=O(exp(exp(polyN))) to withinrelativeprecision  O (N−δ){{\,\mathrm{O}\,}}(N^{-\delta })O(N−δ) O(N-δ) . This comes at the expense of increasing the locality by at most one, and adding an at most poly-sized ancillary system for each coupling; interactions on the ancillary system are geometrically local, and can be translationally invariant. In order to prove this claim, we borrow a technique from high energy physics—where matter fields obtain effective properties (such as mass) from interactions with an exchange particle—and employ a tiling Hamiltonian to discard all cross-terms at higher expansion orders of a Feynman–Dyson series expansion. As an application, we discuss implications for QMA-hardness of theLocal Hamiltonianproblem, and argue that “almost” translational invariance—defined as arbitrarily small relative variations of the strength of the local terms—is as good as non-translational invariance in many of the constructions used throughout Hamiltonian complexity theory. We furthermore show that the choice of geared limit of many-body systems, where e.g. width and height of a lattice are taken to infinity in a specific relation, can have different complexity-theoretic implications: even for translationally invariant models, changing the geared limit can vary the hardness of finding the ground state energy with respect to a given promise gap from computationally trivial, to QMAEXP-, or even BQEXPSPACE-complete.
Application of the adiabatic model of quantum computation requires efficient encoding of the solution to computational problems into the lowest eigenstate of a Hamiltonian that supports universal adiabatic quantum computation. Experimental systems are typically limited to restricted forms of two-body interactions. Therefore, universal adiabatic quantum computation requires a method for approximating quantum many-body Hamiltonians up to arbitrary spectral error using at most two-body interactions. Hamiltonian gadgets, introduced around a decade ago, offer the only current means to address this requirement. Although the applications of Hamiltonian gadgets have steadily grown since their introduction, little progress has been made in overcoming the limitations of the gadgets themselves. In this experimentally motivated theoretical study, we introduce several gadgets which require significantly more realistic control parameters than similar gadgets in the literature. We employ analytical techniques which result in a reduction of the resource scaling as a function of spectral error for the commonly used subdivision, three- to two-body and kkk-body gadgets. Accordingly, our improvements reduce the resource requirements of all proofs and experimental proposals making use of these common gadgets. Next, we numerically optimize these gadgets to illustrate the tightness of our analytical bounds. Finally, we introduce a gadget that simulates a YYYYYY interaction term using Hamiltonians containing only X,Z,XX,ZZ{X,Z,XX,ZZ}X,Z,XX,ZZ terms. Apart from possible implications in a theoretical context, this work could also be useful for a first experimental implementation of these key building blocks by requiring less control precision without introducing extra ancillary qubits.
Perturbative gadgets are a tool to encode part of a Hamiltonian, usually the low-energy subspace, into a different Hamiltonian with favorable properties, for instance, reduced locality. Many constructions of perturbative gadgets have been proposed over the years. Still, all of them are restricted in some ways: Either they apply to some specific classes of Hamiltonians, they involve recursion to reduce locality, or they are limited to studying time evolution under the gadget Hamiltonian, e.g., in the context of adiabatic quantum computing, and thus involve subspace restrictions. In this work, we fill the gap by introducing a versatile universal, nonrecursive, nonadiabatic perturbative gadget construction without subspace restrictions, that encodes an arbitrary many-body Hamiltonian into the low-energy subspace of a three-body Hamiltonian and is therefore applicable to gate-based quantum computing. Our construction requires rkrkrk additional qubits for a kkk-body Hamiltonian comprising rrr terms. Besides a specific gadget construction, we also provide a recipe for constructing similar gadgets, which can be tailored to different properties, which we discuss.
Abstract Continuous-time quantum hardware implementations generally lack the native capability to implement high-order terms that would facilitate efficient compilation of quantum algorithms. This limitation has, in part, motivated the development of perturbative gadgets—multi-qubit constructions used to effect a desired Hamiltonian using engineered low-energy subspaces of a larger system constructed using simpler, usually two-body, primitives. In this work, we demonstrate how a class of non-perturbative gadgets can produce high-order multi-body interactions by taking advantage of the odd–even properties of topological defect subspaces. The simplest example uses domain-wall defects in an effective Ising spin chain with linear connectivity and three-body couplings, alongside three- or five-body driving terms depending on the intended use. We demonstrate a version of a gadget which can perform an encoded bit-flip operation on a minor embedding chain, an important task to mitigate the limitations of quasi-two-dimensional (also sometimes called quasi-planar) topology. Although this will be the main focus of the paper due to conceptual simplicity, there exist systems constructed with only two-body couplings where the boundaries determine whether there are an odd or even number of defects, namely ice-like systems which may yield more complex gadget-like constructions.