Abstract
We investigate the problem of fast-forwarding quantum evolution, whereby the dynamics of certain quantum systems can be simulated with gate complexity that is sublinear in the evolution time. We provide a definition of fast-forwarding that considers the model of quantum computation, the Hamiltonians that induce the evolution, and the properties of the initial states. Our definition accounts for any asymptotic complexity improvement of the general case and we use it to demonstrate fast-forwarding in several quantum systems. In particular, we show that some local spin systems whose Hamiltonians can be taken into block diagonal form using an efficient quantum circuit, such as those that are permutation-invariant, can be exponentially fast-forwarded. We also show that certain classes of positive semidefinite local spin systems, also known as frustration-free, can be polynomially fast-forwarded, provided the initial state is supported on a subspace of sufficiently low energies. Last, we show that all quadratic fermionic systems and number-conserving quadratic bosonic systems can be exponentially fast-forwarded in a model where quantum gates are exponentials of specific fermionic or bosonic operators, respectively. Our results extend the classes of physical Hamiltonians that were previously known to be fast-forwarded, while not necessarily requiring methods that diagonalize the Hamiltonians efficiently. We further develop a connection between fast-forwarding and precise energy measurements that also accounts for polynomial improvements.
Highlights
Only a handful, and the field of fast-forwarding quantum evolution remains largely unexplored
The requirements to achieve exponential fast-forwarding using the Lie-algebra diagonalization approach are satisfied by some important classes of Hamiltonians, such as quadratic fermionic and bosonic Hamiltonians, as we show below
We studied the problem of fast-forwarding quantum evolution in various physical settings and under fairly general conditions, going beyond previous studies [15]
Summary
For a given Hamiltonian H and t > 0, Hamiltonian simulation methods aim at simulating the evolution operator U (t) := e−itH. Known no-fast-forwarding results [4, 15, 16] place a lower bound on the worst-case quantum complexity for simulating classes of Hamiltonians as long as the evolution time satisfies t ≤ T , where T depends on certain problem parameters, such as the number of qubits or spins. These lower bounds are commonly presented as asymptotic scalings. We use A, A , etc., to denote ancillary systems, which include all additional systems required by a particular Hamiltonian simulation approach, such as those needed to implement certain classical computations reversibly
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