Hamiltonian simulation in the interaction picture
In this disclosure, quantum algorithms are presented for simulating Hamiltonian time-evolution e −i(A+B)t in the interaction picture of quantum mechanics on a quantum computer. The interaction picture is a known analytical tool for separating dynamical effects due to trivial free-evolution A from those due to interactions B. This is especially useful when the energy-scale of the trivial component is dominant, but of little interest. Whereas state-of-art simulation algorithms scale with the energy ∥A+B∥≤∥A∥+∥B∥ of the full Hamiltonian, embodiments of the disclosed approach generally scale linearly with the sum of the Hamiltonian coefficients from the low-energy component B and poly-logarithmically with those from A.
1. A method, comprising:
configuring a quantum computer to simulate time-evolution of a quantum system represented by a time-independent Hamiltonian of the form H=A+B, wherein A and B are two non-commuting parts, a free particle theory part A and a part describing interactions B by:
transforming the time-independent Hamiltonian to a time-dependent Hamiltonian, wherein the time-dependent Hamiltonian has the form H(t)=e iAt Be −iAt ;
approximating a truncation and discretization of a Dyson series for the time-dependent Hamiltonian H(t) with a spectral norm α=max t ∥H(t)∥ and average rate-of change
〈
H
.
〉
=
1
t
∫
0
t
H
.
(
s
)
ds
;
and
performing a simulation of the quantum system using the quantum computer by implementing a circuit of quantum gates.
2. The method of claim 1 , wherein the simulation is performed using linear combinations of unitaries.
3. The method of claim 2 , wherein the simulation uses linear combinations of unitaries performed on a diagonally dominant matrix.
4. The method of claim 3 , wherein the simulation using linear combinations of unitaries is performed on diagonally dominant components of the diagonally dominant matrix.
5. The method of claim 1 , wherein the quantum system is modelled by a Hubbard model.
6. The method of claim 1 , wherein the quantum system describes a physical chemical system or molecule.
7. The method of claim 1 , wherein the Hamiltonian is sparse and the simulation uses a state of an auxiliary qubit to encode matrix elements of the Hamiltonian instead of using graph decomposition techniques.
8. The method of claim 1 , further comprising reducing the number of ancillas for quantum simulation of a time-dependent Hamiltonian by avoiding duplication of registers needed to perform the quantum simulation.
9. A computing system, comprising:
a classical computer configured to:
transform a time-independent Hamiltonian of the form H=A+B wherein A and B are two non-commuting parts, a free-particle theory part, A, and a part describing interactions, B, to a time-dependent Hamiltonian in the interaction picture wherein the time-dependent Hamiltonian has the form H(t)=e iAt Be −iAt ;
approximate a truncation and discretization of a Dyson series for the time-dependent Hamiltonian, H(t), with a spectral-norm α=max t ∥H(t)∥ and an average rate-of-change and
〈
H
.
〉
=
1
t
∫
0
t
H
.
(
s
)
ds
;
compile a circuit of quantum gates representing the approximated truncated and discretized Dyson series; and
a quantum computer configured to perform a simulation of the quantum system by implementing the circuit of quantum gates.
10. A method, comprising:
implementing a quantum algorithm on a quantum computer for simulating a general sparse time-dependent quantum system represented by a time-dependent Hamiltonian H(t), wherein the quantum algorithm does not use graph decomposition techniques by:
approximating a truncation and discretization of a Dyson series for the time-dependent Hamiltonian, H(t), with a spectral-norm α=max t ∥h(t)∥ and average rate-of-change
〈
H
.
〉
=
1
t
∫
0
t
H
.
(
s
)
ds
;
compiling a circuit of quantum gates representing the approximated truncated and discretized Dyson series; and
implementing the quantum algorithm on the quantum computer by implementing the circuit of quantum gates.
11. The method of claim 10 , wherein the time-dependent Hamiltonian used in the simulation is represented in the interaction picture.
12. The method of claim 10 , wherein the simulation uses linear combinations of unitaries performed on a diagonally dominant matrix.
13. The method of claim 10 , wherein the quantum system is modelled by a Hubbard model.
14. The method of claim 10 , wherein the quantum system describes a physical chemical system, or a molecule.
15. The method of claim 10 , further comprising reducing the number of ancillas used to index a time for the simulation by avoiding duplication of registers needed to index a time for the simulation.