Efficient Quantum Simulation

Quantum simulation, the idea that you can simulate a quantum system using another quantum system, is thought to be one of the main applications of quantum computation. Feynman was the first to suggest the this might be useful, given the difficulties of performing quantum simulations on classical machines [1]. Simulating the time evolution of an arbitrary quantum system is intractable for a classical machine [2]. In other words, simulating a quantum system on a classical computer scales exponentially with the complexity of the system. We can see this by considering a collection of n two-level systems (qubits). To record the state of the system we need to store 2 complex numbers. To calculate the time evolution of this system, we need to exponentiate a 2 × 2 matrix. For a relatively modest number of qubits, this number is very large (e.g. n = 40 is a matrix of ∼ 10 entries). A ‘back of the envelope’ calculation by Brown et al. [3] shows that for a modest n = 27, to store the complex numbers needed to define a quantum state would require ∼ 1gb of memory. One of the largest classical simulations of a quantum system to date required 1tb of memory, 4096 processors and involved the simulation of 36 qubits [4]. It is unlikely that there will ever be an efficient classical algorithm for simulating the dynamics of quantum systems. The problem is BQP-complete, and so an efficient classical algorithm for quantum simulation would mean that any problem that is efficient for a quantum computer could be done efficiently on a classical one (e.g. Shor’s factorisation algorithm) [5]. There are two general approaches that can be taken with respect to quantum simulation. The first, often referred to as ‘analogue quantum simulation’, is a simulation of a quantum system by use of another system with similar dynamics. In this approach, the Hamiltonian of the system you are trying to simulate, Hsys, is mapped onto the Hamiltonian of the simulator, Hsim, which you have more control of. In essence the simulator system is used to mimic the first