Hamiltonian gadgets with reduced resource requirements

Abstract

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 2-body interactions. Therefore, universal adiabatic quantum computation requires a method for approximating quantum many-body Hamiltonians up to arbitrary spectral error using at most 2-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, 3-to 2-body and k-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 new gadgets to illustrate the tightness of our analytical bounds. Finally, we introduce a new gadget that simulates a Y Y interaction term using Hamiltonians containing only {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.Although adiabatic quantum computation is known to be a universal model of quantum computation [1][2][3][4][5] and hence, in principle equivalent to the circuit model, the mappings between an adiabatic process and an arbitrary quantum circuit require significant overhead. Currently the approaches to universal adiabatic quantum computation require implementing multiple higher order and non-commuting interactions by means of perturbative gadgets [4]. Such gadgets arose in early work on quantum complexity theory and the resources required for their implementation are the subject of this study.Early work by Kitaev et al. [6] established that an otherwise arbitrary Hamiltonian restricted to have at most 5-body interactions has a ground state energy problem which is complete for the quantum analog of the complexity class NP (QMA-complete). Reducing the locality of the Hamiltonians from 5-body down to 2-body remained an open problem for a number of years. In their 2004 proof that 2-local Hamiltonian is QMA-Complete, Kempe, Kitaev and Regev formalized the idea of a perturbative gadget, which finally accomplished this task [7]. Oliveira and Terhal further reduced the problem, showing completeness when otherwise arbitrary 2-body Hamiltonians were restricted to act on a square lattice [3]....

Exploiting Locality in Quantum Computation for Quantum Chemistry

“…Though known to be polynomial, it is extremely difficult to predict exactly how ∆ depends on M 2 as applying gadgets to terms "in parallel" leads to "cross-gadget contamination" which contributes at high orders in the perturbative expansion of the self-energy used to analyze these gadgets [55]. Without a significantly deeper analysis, we can only conclude that, ∆ = Ω poly (M ) β OEI max M 2/3 log(M ) .…”

“…A positive answer to this conjecture would allow us to embed molecular electronic structure Hamiltonians without needing large spectral gaps. For entirely diagonal Hamiltonians, such gadgets are well known in the literature [56,57] but fail when terms do not commute [55]. Exact reductions have also been achieved for certain Hamiltonians.…”

“…In the supplementary material presented in a later paper analyzing the scaling of gadget constructions [55], it is shown that for bit-flip gadgets, λ k+1 /∆ k = O ( ) and…”

Adiabatic state preparation study of methylene

“…Therefore, when implementing AQC experimentally, one has to transform the Hamiltonian (2) of a quantum register to contain at most 2-qubit interactions. One of the possibilities of transforming general Hamiltonians containing non-commuting k-qubit terms to 2-qubit terms are methods of perturbative gadgets [43,[47][48][49]. We will use this approach in Section IV to transform 4-qubit terms to 2-qubit terms in case of the small CH 2 experimental proposal.…”

“…We have used the perturbative gadgets technique [48] to transform the 4-qubit terms to 2-qubit terms at the cost of 16 ancilla qubits. For a detailed comparison of different types of gadgets with the emphasis on an experimental accessibility, see [49]. Our proposal thus requires a total number of 20 qubits.…”