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arXiv:quant-ph/0403082 · Physical Review A

Gate simulation and lower bounds on the simulation time

Robert Zeier, Markus Grassl, Thomas Beth

Published Sep 22, 2004• 26 citations
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Abstract

Unitary operations are the building blocks of quantum programs. Our task is to design efficient or optimal implementations of these unitary operations by employing the intrinsic physical resources of a given $n$-qubit system. The most common versions of this task are known as Hamiltonian simulation and gate simulation, where Hamiltonian simulation can be seen as an infinitesimal version of the general task of gate simulation. We present a Lie-theoretic approach to Hamiltonian simulation and gate simulation. From this, we derive lower bounds on the time complexity in the $n$-qubit case, generalizing known results to both even and odd $n$. To achieve this we develop a generalization of the so-called magic basis for two-qubits. As a corollary, we note a connection to entanglement measures of concurrence-type.

DOI
10.1103/physreva.70.032319

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Unitary operations are the building blocks of quantum programs. Our task is to design efficient or optimal implementations of these unitary operations by employing the intrinsic physical resources of a given nnn-qubit system. The most common versions of this task are known as Hamiltonian simulation and gate simulation, where Hamiltonian simulation can be seen as an infinitesimal version of the general task of gate simulation. We present a Lie-theoretic approach to Hamiltonian simulation and gate simulation. From this, we derive lower bounds on the time complexity in the nnn-qubit case, generalizing known results to both even and odd nnn. To achieve this we develop a generalization of the so-called magic basis for two-qubits. As a corollary, we note a connection to entanglement measures of concurrence-type.
We present general methods for simulating black-box Hamiltonians using quantum walks. These techniques have two main applications: simulating sparse Hamiltonians and implementing black-box unitary operations. In particular, we give the best known simulation of sparse Hamiltonians with constant precision. Our method has complexity linear in both the sparseness $\sparseness$ (the maximum number of nonzero elements in a column) and the evolution time ttt, whereas previous methods had complexity scaling as $\sparseness^4$ and were superlinear in ttt. We also consider the task of implementing an arbitrary unitary operation given a black-box description of its matrix elements. Whereas standard methods for performing an explicitly specified $\UN\times\UN$ unitary operation use $\tilde O(\UN^2)$ elementary gates, we show that a black-box unitary can be performed with bounded error using $O(\UN^{2/3} (\log\log \UN)^{4/3})$ queries to its matrix elements. In fact, except for pathological cases, it appears that most unitaries can be performed with only $\tilde O(\sqrt{\UN})$ queries, which is optimal.
We formulate and investigate the simplest version of time-optimal quantum computation theory (TO-QCT), where the computation time is defined by the physical one and the Hamiltonian contains only one- and two-qubit interactions. This version of TO-QCT is also considered as optimality by sub-Riemannian geodesic length. The work has two aims: One is to develop a TO-QCT itself based on a physically natural concept of time, and the other is to pursue the possibility of using TO-QCT as a tool to estimate the complexity in conventional gate-optimal quantum computation theory (GO-QCT). In particular, we investigate to what extent is true the following statement: Time complexity is polynomial in the number of qubits if and only if gate complexity is also. In the analysis, we relate TO-QCT and optimal control theory (OCT) through fidelity-optimal computation theory (FO-QCT); FO-QCT is equivalent to TO-QCT in the limit of unit optimal fidelity, while it is formally similar to OCT. We then develop an efficient numerical scheme for FO-QCT by modifying Krotov's method in OCT, which has a monotonic convergence property. We implemented the scheme and obtained solutions of FO-QCT and of TO-QCT for the quantum Fourier transform and a unitary operator that does not have an apparent symmetry. The former has a polynomial gate complexity and the latter is expected to have an exponential one which is based on the fact that a series of generic unitary operators has an exponential gate complexity. The time complexity for the former is found to be linear in the number of qubits, which is understood naturally by the existence of an upper bound. The time complexity for the latter is exponential in the number of qubits. Thus, both the targets seem to be examples satisfyng the preceding statement. The typical characteristics of the optimal Hamiltonians are symmetry under time reversal and constancy of one-qubit operation, which are mathematically shown to hold in fairly general situations.
Digital-analog is a quantum computational paradigm that employs the natural interaction Hamiltonian of a system as the entangling resource, combined with single qubit gates, to implement universal quantum operations. As in the case of its digital gate-based counterpart, designing digital-analog circuits that employ optimal quantum resources often requires an exceedingly large classical computational time. In this work we find a suboptimal solution to this exponentially large problem, showing that it can be solved within polynomial computational time. In particular, we provide an exact solution for the problem of expressing arbitrary two-body Hamiltonians as the sum of local unitary transformations of an arbitrary Ising Hamiltonian, with the total number of required terms being at most quadratic in system size. This allows us to design a digital-analog simulation protocol that avoids employing numerical optimization over a large parameter space at the preprocessing stage, minimizing computational resources and allowing for further scaling.
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.
For Hamiltonian H=∑jhjH = \sum_j h_jH=∑j​hj​, we prove asymptotically tight lower bounds on the gate and query complexities of simulating time evolution on a quantum computer. Our bounds hold for arbitrary term norms ∥hj∥\|h_j\|∥hj​∥, time ttt, and trace-distance error εεε. The matching upper bound (known as composite qDRIFT) consists of high-order Trotterization of the large terms and a randomized first-order Trotterization of the small terms. Unlike prior work that chooses worst-case ∥hj∥\|h_j\|∥hj​∥ to encode the computation of parity or other Boolean functions in time evolution, our proof is elementary and based on a local, bounded-degree classical Hamiltonian. Our work suggests that for many physical systems (e.g., power-law interactions), gate count must scale polynomially in 1/ε1/ε1/ε, contrary to the complexity suggested by counting coherent oracle queries such as those in the block-encoding model.