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arXiv:quant-ph/0009094 · Physical Review A

Complexity measure for continuous-time quantum algorithms

Dominik Janzing, T. Beth

Published Jul 2, 2001• 7 citations
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Abstract

We consider unitary dynamical evolutions on n qubits caused by time-dependent pair-interaction Hamiltonians and show that the running time of a parallelized two-qubit gate network simulating the evolution is given by the time integral over the chromatic index of the interaction graph. This defines complexity measures of continuous and discrete quantum algorithms, which are in exact one-to-one correspondence. We prove a lower bound on the complexity of those multiparticle states, which show quantum superpositions on the macroscopic scale.

DOI
10.1103/physreva.64.022301

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We consider unitary dynamical evolutions on n qubits caused by time-dependent pair-interaction Hamiltonians and show that the running time of a parallelized two-qubit gate network simulating the evolution is given by the time integral over the chromatic index of the interaction graph. This defines complexity measures of continuous and discrete quantum algorithms, which are in exact one-to-one correspondence. We prove a lower bound on the complexity of those multiparticle states, which show quantum superpositions on the macroscopic scale.
We consider a quantum computer consisting of n spins with an arbitrary but fixed pair-interaction Hamiltonian and describe how to simulate other pair-interactions by interspersing the natural time evolution with fast local transformations. Calculating the minimal time overhead of such a simulation leads to a convex optimization problem. Lower and upper bounds on the minimal time overhead are derived in terms of chromatic indices of interaction graphs and spectral majorization criteria. These results classify Hamiltonians with respect to their computational power. For a specific Hamiltonian, namely \sigma_z\otimes\sigma_z-interactions between all spins, the optimization is mathematically equivalent to a separability problem of n-qubit density matrices. We compare the complexity defined by such a quantum computer with the usual gate complexity.
The most popular model of quantum computation is the quantum circuit model using single and two qubit gates as elementary transformations. Definitions of quantum complexity usually refer to this model. In contrast, we consider the physical interactions among the qubits as the basic computational resource that allows to carry out complex transformations on the quantum register. One method to compare the computational power of different interactions is to examine the complexity of the so-called mutual simulation of interaction Hamiltonians. The physical interaction can simulate other interactions by interspersing the natural time evolution with external control operations. In previous papers we have considered mutual simulation of n-partite pair-interaction Hamiltonians. We have focussed on the running time overhead of general simulations, while considering the required number of time steps only for special cases (decoupling and time-reversal). These two complexity measures differ significantly. Here we derive lower bounds on the number of time steps for simulations of general interaction graphs. In particular, the simulation of interaction graphs with irrational spectrum requires at least n steps. We discuss as examples graphs that correspond to graph codes and nearest neighbor interactions in 1- and 2-dimensional lattices. In the latter case the lower bounds are almost tight.
We study the problem of simulating the time evolution of a lattice Hamiltonian, where the qubits are laid out on a lattice and the Hamiltonian only includes geometrically local interactions (i.e., a qubit may only interact with qubits in its vicinity). This class of Hamiltonians is very general and is believed to capture fundamental interactions of physics. Our algorithm simulates the time evolution of such a Hamiltonian on nnn qubits for time TTT up to error ϵ\epsilonϵ using O(nTpolylog(nT/ϵ)){\mathcal O}( nT {polylog} (nT/\epsilon))O(nTpolylog(nT/ϵ)) gates with depth O(Tpolylog(nT/ϵ)){\mathcal O}(T { polylog} (nT/\epsilon))O(Tpolylog(nT/ϵ)). Our algorithm is the first simulation algorithm that achieves gate cost quasilinear in nTnTnT and polylogarithmic in 1/ϵ1/\epsilon1/ϵ. Our algorithm also readily generalizes to time-dependent Hamiltonians and yields an algorithm with similar gate count for any piecewise slowly varying time-dependent bounded local Hamiltonian. We also prove a matching lower bound on the gate count of such a simulation, showing that any quantum algorithm that can simulate a piecewise constant bounded local Hamiltonian in one dimension to constant error requires Ω~(nT){\widetilde{\Omega}}(nT)Ω(nT) gates in the worst case. The lower bound holds even if we only require the output state to be correct on local measurements. To the best of our knowledge, this is the first nontrivial lower bound on the gate complexity of the simulation problem. Our algorithm is based on a decomposition of the time-evolution unitary into a product of small unitaries using Lieb--Robinson bounds. In the appendix, we prove a Lieb--Robinson bound tailored to Hamiltonians with small commutators between local terms, giving zero Lieb--Robinson velocity in the limit of commuting Hamiltonians. This improves the performance of our algorithm when the Hamiltonian is close to commuting.
Entanglement is one of the physical properties of quantum systems responsible for the computational hardness of simulating quantum systems. But while the runtime of specific algorithms, notably tensor network algorithms, explicitly depends on the amount of entanglement in the system, it is unknown whether this connection runs deeper and entanglement can also cause inherent, algorithm-independent complexity. In this Letter, we quantitatively connect the entanglement present in certain quantum systems to the computational complexity of simulating those systems. Moreover, we completely characterize the entanglement and complexity as a function of a system parameter. Specifically, we consider the task of simulating single-qubit measurements of k-regular graph states on n qubits. We show that, as the regularity parameter is increased from 1 to n-1, there is a sharp transition from an easy regime with low entanglement to a hard regime with high entanglement at k=3, and a transition back to easy and low entanglement at k=n-3. As a key technical result, we prove a duality for the simulation complexity of regular graph states between low and high regularity.
We give a quantum algorithm for Lindbladian simulation given a block encoding of the Hamiltonian HHH and a projected unitary encoding of the stacked jump operator B=∑k=1m∣k⟩⊗LkB=\sum_{k=1}^m \lvert k\rangle\otimes L_kB=∑k=1m​∣k⟩⊗Lk​, with normalization factors αHα_HαH​ and αBα_BαB​, respectively. For evolution time ttt, set τ=(αH+αB2)tτ=(α_H+α_B^2)tτ=(αH​+αB2​)t. The algorithm approximates the evolution channel to diamond-norm error ε\varepsilonε using O ⁣(τ+log⁡(1/ε)log⁡ ⁣(e+log⁡(1/ε)/τ))O\!\left(τ+\frac{\log(1/\varepsilon)}{\log\!\left(e+\log(1/\varepsilon)/τ\right)}\right)O(τ+log(e+log(1/ε)/τ)log(1/ε)​) oracle queries, matching the query lower bound for Hamiltonian simulation. The number of additional one- and two-qubit gates is linear in the query complexity up to polylogarithmic factors. The query- and gate-complexity bounds extend to Lipschitz-continuous time-dependent Lindbladians under coherent time-indexed oracle access. Our construction uses a one-query transducer that implements a product of rational approximations to short-time evolution when supplied with a catalyst. We bound the error from omitting the catalyst by exploiting orthogonality between different sequences of Kraus labels. The gate implementation combines a compressed Kraus-label representation, which stores only the positions and values of the nonzero labels, with the rotation factorization of Chen et al.
We study the problem of simulating the time evolution of a lattice Hamiltonian, where the qubits are laid out on a lattice and the Hamiltonian only includes geometrically local interactions (i.e., a qubit may only interact with qubits in its vicinity). This class of Hamiltonians is very general and encompasses all physically reasonable Hamiltonians. Our algorithm simulates the time evolution of such a Hamiltonian on n qubits for time T up to error ε using O(T polylog(nT/ε)) gates with depth O(T polylog(nT/ε)). Our algorithm is the first simulation algorithm that achieves gate cost quasilinear in nT and polylogarithmic in 1/ε. Our algorithm also readily generalizes to time-dependent Hamiltonians and yields an algorithm with similar gate count for any piecewise slowly varying time-dependent bounded local Hamiltonian. We also prove a matching lower bound on the gate count of such a simulation, showing that any quantum algorithm that can simulate a piecewise constant bounded local Hamiltonian in one dimension to constant error requires (nT) gates in the worst case. The lower bound holds even if we only require the output state to be correct on local measurements. To our best knowledge, this is the first nontrivial lower bound on the gate complexity of the simulation problem. Our algorithm is based on a decomposition of the time-evolution unitary into a product of small unitaries using Lieb-Robinson bounds. In the appendix, we prove a Lieb-Robinson bound tailored to Hamiltonians with small commutators between local terms, giving zero Lieb-Robinson velocity in the limit of commuting Hamiltonians. This improves the performance of our algorithm when the Hamiltonian is close to commuting.