Analysis and limitations of modified circuit-to-Hamiltonian constructions

We present a collection of results about the clock in Feynman's computer construction and Kitaev's Local Hamiltonian problem. First, by analyzing the spectra of quantum walks on a line with varying endpoint terms, we find a better lower bound on the gap of the Feynman Hamiltonian, which translates into a less strict promise gap requirement for the QMA-complete Local Hamiltonian problem. We also translate this result into the language of adiabatic quantum computation. Second, introducing an idling clock construction with a large state space but fast Cesaro mixing, we provide a way for achieving an arbitrarily high success probability of computation with Feynman's computer with only a logarithmic increase in the number of clock qubits. Finally, we tune and thus improve the costs (locality, gap scaling) of implementing a (pulse) clock with a single excitation.

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“…Note that Bausch & Crosson have independently found a comparable (tight Θ(N −2 ) promise gap) result in [4].…”

Section: A New Promise Gap Bound For Kitaev's Qma-complete Local Hamimentioning

confidence: 79%

Clocks in Feynman's computer and Kitaev's local Hamiltonian: Bias, gaps, idling, and pulse tuning

2018

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“…As a short digression, we emphasize that this result is weaker than it seems: QMAhardness constructions are commonly given with a promise gap that scales as ∝ 1/T 2 in the runtime T of the embedded computation (see [BC18], and note the link to our bound state Hamiltonian in section 2.3). For QMA (QMA EXP ), the runtime is T = poly n for a system size n. In order to lift the promise gap arbitrarily close to constant in the system size, it suffices to add a polynomially-sized non-interacting slack of size n = poly n; if we express T in n , we can thus get a scaling T = n 1/a , for some arbitrarily large a > 0.…”

Section: The Local Hamiltonian Problemmentioning

confidence: 93%

Perturbation Gadgets: Arbitrary Energy Scales from a Single Strong Interaction

Bausch

2019

Ann. Henri Poincaré

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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 appears k-body and approximates a target Hamiltonian to within precision . One caveat of existing constructions is that the interaction strength generally scales exponentially in the locality of the terms to be approximated, i.e. as Ω(1/ k ); if = 1/poly n in the system size n, as is necessary for e.g. QMA-hardness constructions, the energy differences become highly unphysical.In this work we propose a many-body Hamiltonian construction which introduces only a single separate energy scale of order Θ(1/N 2+δ ), for a small parameter δ > 0, and for N terms in the target Hamiltonian-i.e. all local terms of the simulator have either this norm, or one of O(1). In its low-energy subspace, we can approximate any normalized target Hamiltonian H t = N i=1 h i with norm ratios r = h i 2 / h j 2 = O(exp(exp(poly n))) to within relative precision O(N −δ ). This comes at the expense of increasing the locality by at most one, and adding an at most poly-sized ancilliary system for each coupling; the ancillas being qutrits for exponential scaling, and qudits for doubly exponential r; the interactions on the ancilliary 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 a tiling Hamiltonian to drop all cross terms at higher expansion orders, which simplifies the analysis of a traditional Feynman-Dyson series expansion.As an application, we discuss implications for QMA-hardness of the L H problem, 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 QMA EXP -, or even BQEXPSPACE-complete.

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“…The register C, called the clock register, indicates how many gates have been applied to the all zeroes state, which is stored in register S (called the state register) containing an initial state |ψ and ancillas. Although this state has only a 1/(T + 1) fidelity with the output of the circuit, the standard technique for increasing the overlap to be inverse polynomially close to 1 is to pad the end of the circuit with identity gates (for recent work on more efficient methods for biasing the history state towards its endpoints, see [BC18,CLN18]). This technique allows history states to capture approximate versions of QECC that have efficient encoding circuits.…”

Section: Description Of the Code Hamiltonianmentioning

confidence: 99%

“…In this section, we describe another closely related construction of approximate LDPC codes, which is based on using the standard Feynman-Kitaev construction with a global clock as well as recently-introduced variants that increase the overlap of the history state with the beginning and end of the computation [BC18,CLN18]. The primary advantage of this version of the construction is the significantly simpler analysis of the spectral gap, even in the presence of nonuniform weight distributions on the time steps of the computation.…”

Section: Good Approximate Qldpc From Weighted Fk Hamiltoniansmentioning

confidence: 99%

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Good approximate quantum LDPC codes from spacetime circuit Hamiltonians

Bohdanowicz

Crosson

Nirkhe

et al. 2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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We study approximate quantum low-density parity-check (QLDPC) codes, which are approximate quantum error-correcting codes specified as the ground space of a frustration-free local Hamiltonian, whose terms do not necessarily commute. Such codes generalize stabilizer QLDPC codes, which are exact quantum error-correcting codes with sparse, low-weight stabilizer generators (i.e. each stabilizer generator acts on a few qubits, and each qubit participates in a few stabilizer generators). Our investigation is motivated by an important question in Hamiltonian complexity and quantum coding theory: do stabilizer QLDPC codes with constant rate, linear distance, and constant-weight stabilizers exist?We show that obtaining such optimal scaling of parameters (modulo polylogarithmic corrections) is possible if we go beyond stabilizer codes: we prove the existence of a family of [[N, k, d, ε]] approximate QLDPC codes that encode k = Ω(N) logical qubits into N physical qubits with distance d = Ω(N) and approximation infidelity ε = O(1/ polylog(N)). The code space is stabilized by a set of 10-local noncommuting projectors, with each physical qubit only participating in O(polylog N) projectors. We prove the existence of an efficient encoding map and show that the spectral gap of the code Hamiltonian scales as Ω(N −3.09 ). We also show that arbitrary Pauli errors can be locally detected by circuits of polylogarithmic depth.Our family of approximate QLDPC codes is based on applying a recent connection between circuit Hamiltonians and approximate quantum codes (Nirkhe, et al., ICALP 2018) to a result showing that random Clifford circuits of polylogarithmic depth yield asymptotically good quantum codes (Brown and Fawzi, ISIT 2013). Then, in order to obtain a code with sparse checks and strong detection of local errors, we use a spacetime circuit Hamiltonian construction in order to take advantage of the parallelism of the Brown-Fawzi circuits.The analysis of the spectral gap of the code Hamiltonian is the main technical contribution of this work. We show that for any depth D quantum circuit on n qubits there is an associated spacetime circuit-to-Hamiltonian construction with spectral gap Ω(n −3.09 D −2 log −6 (n)). To lower bound this gap we use a Markov chain decomposition method to divide the state space of partially completed circuit configurations into overlapping subsets corresponding to uniform circuit segments of depth log n, which are based on bitonic sorting circuits. We use the combinatorial properties of these circuit configurations to show rapid mixing between the subsets, and within the subsets we develop a novel isomorphism between the local update Markov chain on bitonic circuit configurations and the edge-flip Markov chain on equal-area dyadic tilings, whose mixing time was recently shown to be polynomial (Cannon, Levin, and Stauffer, RANDOM 2017). Previous lower bounds on the spectral gap of spacetime circuit Hamiltonians have all been based on a connection to exactly solvable quantum spin chains and applied only to ...

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Analysis and limitations of modified circuit-to-Hamiltonian constructions

Cited by 26 publications

References 29 publications

Clocks in Feynman's computer and Kitaev's local Hamiltonian: Bias, gaps, idling, and pulse tuning

Clocks in Feynman's computer and Kitaev's local Hamiltonian: Bias, gaps, idling, and pulse tuning

Perturbation Gadgets: Arbitrary Energy Scales from a Single Strong Interaction

Good approximate quantum LDPC codes from spacetime circuit Hamiltonians

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