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Analysis and limitations of modified circuit-to-Hamiltonian constructions

Johannes Bausch, Elizabeth Crosson

Published Sep 19, 2018• 16 citations
Publisher

Abstract

Feynman's circuit-to-Hamiltonian construction connects quantum computation and ground states of many-body quantum systems. Kitaev applied this construction to demonstrate QMA-completeness of the local Hamiltonian problem, and Aharanov et al. used it to show the equivalence of adiabatic computation and the quantum circuit model. In this work, we analyze the low energy properties of a class of modified circuit Hamiltonians, which include features like complex weights and branching transitions. For history states with linear clocks and complex weights, we develop a method for modifying the circuit propagation Hamiltonian to implement any desired distribution over the time steps of the circuit in a frustration-free ground state, and show that this can be used to obtain a constant output probability for universal adiabatic computation while retaining the Ω ( T − 2 ) scaling of the spectral gap, and without any additional overhead in terms of numbers of qubits.Furthermore, we establish limits on the increase in the ground energy due to input and output penalty terms for modified tridiagonal clocks with non-uniform distributions on the time steps by proving a tight O ( T − 2 ) upper bound on the product of the spectral gap and ground state overlap with the endpoints of the computation. Using variational techniques which go beyond the Ω ( T − 3 ) scaling that follows from the usual geometrical lemma, we prove that the standard Feynman-Kitaev Hamiltonian already saturates this bound. We review the formalism of unitary labeled graphs which replace the usual linear clock by graphs that allow branching and loops, and we extend the O ( T − 2 ) bound from linear clocks to this more general setting. In order to achieve this, we apply Chebyshev polynomials to generalize an upper bound on the spectral gap in terms of the graph diameter to the context of arbitrary Hermitian matrices.

DOI
10.17863/cam.44094

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Feynman's circuit-to-Hamiltonian construction connects quantum computation and ground states of many-body quantum systems. Kitaev applied this construction to demonstrate QMA-completeness of the local Hamiltonian problem, and Aharanov et al. used it to show the equivalence of adiabatic computation and the quantum circuit model. In this work, we analyze the low energy properties of a class of modified circuit Hamiltonians, which include features like complex weights and branching transitions. For history states with linear clocks and complex weights, we develop a method for modifying the circuit propagation Hamiltonian to implement any desired distribution over the time steps of the circuit in a frustration-free ground state, and show that this can be used to obtain a constant output probability for universal adiabatic computation while retaining the Ω ( T − 2 ) scaling of the spectral gap, and without any additional overhead in terms of numbers of qubits.Furthermore, we establish limits on the increase in the ground energy due to input and output penalty terms for modified tridiagonal clocks with non-uniform distributions on the time steps by proving a tight O ( T − 2 ) upper bound on the product of the spectral gap and ground state overlap with the endpoints of the computation. Using variational techniques which go beyond the Ω ( T − 3 ) scaling that follows from the usual geometrical lemma, we prove that the standard Feynman-Kitaev Hamiltonian already saturates this bound. We review the formalism of unitary labeled graphs which replace the usual linear clock by graphs that allow branching and loops, and we extend the O ( T − 2 ) bound from linear clocks to this more general setting. In order to achieve this, we apply Chebyshev polynomials to generalize an upper bound on the spectral gap in terms of the graph diameter to the context of arbitrary Hermitian matrices.
Feynman's circuit-to-Hamiltonian construction connects quantum computation and ground states of many-body quantum systems. Kitaev applied this construction to demonstrate QMA-completeness of the local Hamiltonian problem, and Aharanov et al. used it to show the equivalence of adiabatic computation and the quantum circuit model. In this work, we analyze the low energy properties of a class of modified circuit Hamiltonians, which include features like complex weights and branching transitions. For history states with linear clocks and complex weights, we develop a method for modifying the circuit propagation Hamiltonian to implement any desired distribution over the time steps of the circuit in a frustration-free ground state, and show that this can be used to obtain a constant output probability for universal adiabatic computation while retaining the Ω(T−2) scaling of the spectral gap, and without any additional overhead in terms of numbers of qubits.Furthermore, we establish limits on the increase in the ground energy due to input and output penalty terms for modified tridiagonal clocks with non-uniform distributions on the time steps by proving a tight O(T−2) upper bound on the product of the spectral gap and ground state overlap with the endpoints of the computation. Using variational techniques which go beyond the Ω(T−3) scaling that follows from the usual geometrical lemma, we prove that the standard Feynman-Kitaev Hamiltonian already saturates this bound. We review the formalism of unitary labeled graphs which replace the usual linear clock by graphs that allow branching and loops, and we extend the O(T−2) bound from linear clocks to this more general setting. In order to achieve this, we apply Chebyshev polynomials to generalize an upper bound on the spectral gap in terms of the graph diameter to the context of arbitrary Hermitian matrices.
The time or cost of simulating a quantum circuit by adiabatic evolution is determined by the spectral gap of the Hamiltonians involved in the simulation. In "standard" constructions based on Feynman's Hamiltonian, such a gap decreases polynomially with the number of gates in the circuit, L. Because a larger gap implies a smaller cost, we study the limits of spectral gap amplification in this context. We show that, under some assumptions on the ground states and the cost of evolving with the Hamiltonians (which apply to the standard constructions), an upper bound on the gap of the order 1/L follows. In addition, if the Hamiltonians satisfy a frustration-free property, the upper bound is of the order 1/L2. Our proofs use recent results on adiabatic state transformations, spectral gap amplification, and the simulation of continuous-time quantum query algorithms. They also consider a reduction from the unstructured search problem, whose lower bound in the oracle cost translates into the upper bounds in the gaps. The impact of our results is that improving the gap beyond that of standard constructions (i.e. 1/L2), if possible, is challenging.
The quantum adiabatic theorem ensures that a slowly changing system, initially prepared in its ground state, will evolve to its final ground state with arbitrary precision. As a first result this thesis extends the original theorem to projection operators keeping the statement valid for Hamiltonians with degenerate ground spaces. Yet the main focus of this work lies in studying the efficiency of quantum circuit simulations by adabatic quantum computation. The standard Hamiltonian construction by Kitaev is based on a path graph reflecting the LLL computation steps and influencing the scaling of the necessary evolution time by its spectral gap of O(1L2)\mathcal{O}\left(\frac{1}{L^2}\right)O(L21​). Aspiring to an improved running time we generalize Kitaev's Hamiltonian to so-called standard graph Hamiltonians based on graph families with a different spectral gap. In this generalized construction the first two time derivatives of the Hamiltonian and the fraction of initial vertices appear as additional parameters of running time. In a first step the time derivatives can be proven to be constant. Expansion results from spectral graph theory however impose a trade-off between the spectral gap and the fraction of initial vertices as well as the fraction of final vertices which corresponds to the probability for obtaining the correct computational output. Graphs with spectral gap O(1Lk)\mathcal{O}\left(\frac{1}{L^k}\right)O(Lk1​), k<2k<2k<2, turn out to contradict very likely, graphs with k<1k<1k<1 even for sure at least one of the efficiency criterias for running time, output probability or Hamiltonian implementation. The above results may also be obstacles for a possible quantum PCP-theorem in complexity theory claiming the local Hamiltonian problem with constant gap to be QMA-complete since the very same Kitaev Hamiltonian is constructed in the QMA-hardness proof for the Local Hamiltonian problem.
The circuit-to-Hamiltonian construction has found widespread use within the field of Hamiltonian complexity, particularly for proving QMA-hardness results. In this work we examine the ground state energies of the Hamiltonian for standard clock constructions and those which require dynamic initialisation. We put exponentially tight bounds on these ground state energies and also determine improved scaling bounds in the case where there is a constant probability of the computation being rejected. Furthermore, we prove a collection of results concerning the low-energy subspace of quantum walks on a line with energy penalties appearing at any point along the walk and introduce some general tools that may be useful for such analyses.
All Hamiltonian complexity results to date have been proven by constructing a local Hamiltonian whose ground state -- or at least some low-energy state -- is a "computational history state", encoding a quantum computation as a superposition over the history of the computation. We prove that all history-state Hamiltonians must be critical. More precisely, for any circuit-to-Hamiltonian mapping that maps quantum circuits to local Hamiltonians with low-energy history states, there is an increasing sequence of circuits that maps to a growing sequence of Hamiltonians with spectral gap closing at least as fast as O(1/n) with the number of qudits n in the circuit. This result holds for very general notions of history state, and also extends to quasi-local Hamiltonians with exponentially-decaying interactions. This suggests that QMA-hardness for gapped Hamiltonians (and also BQP-completeness of adiabatic quantum computation with constant gap) either require techniques beyond history state constructions. Or gapped Hamiltonians cannot be QMA-hard (respectively, BQP-complete).
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 end-point 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 quantum-Merlin-Arthur--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 and gap scaling) of implementing a (pulse) clock with a single excitation.