Quantum Computer with Improved Quantum Optimization by Exploiting Marginal Data

Abstract

A quantum optimization system and method estimate, on a classical computer and for a quantum state, an expectation value of a Hamiltonian, expressible as a linear combination of observables, based on expectation values of the observables; and transform, on the classical computer, one or both of the Hamiltonian and the quantum state to reduce the expectation value of the Hamiltonian.

What is claimed is: 
     
         1 . A quantum optimization method, comprising:
 estimating, on a classical computer and for a quantum state, an expectation value of a Hamiltonian, expressible as a linear combination of observables, based on expectation values of the observables; and   transforming, on the classical computer, one or both of the Hamiltonian and the quantum state to reduce the expectation value of the Hamiltonian.   
     
     
         2 . The quantum optimization method of  claim 1 , further comprising measuring the expectation value of each of the observables on a quantum computer by:
 generating the quantum state on the quantum computer; and   measuring, on the quantum computer, said each of the observables for the quantum state.   
     
     
         3 . The quantum optimization method of  claim 2 , wherein said generating the quantum state includes generating the quantum state with a parametrized quantum circuit programmable via one or more circuit parameters. 
     
     
         4 . The quantum optimization method of  claim 3 , further comprising updating the one or more circuit parameters such that the parametrized quantum circuit outputs an updated quantum state that better approximates a ground state of the Hamiltonian. 
     
     
         5 . The quantum optimization method of  claim 4 , further comprising repeating:
 said generating the quantum state with the parametrized quantum circuit;   said measuring each of the observables for the quantum state;   said transforming one or both of the Hamiltonian and the quantum state;   updating the Hamiltonian based on said transforming; and   said updating the one or more circuit parameters;   until the one or more circuit parameters have converged.   
     
     
         6 . The quantum optimization method of  claim 1 , wherein said transforming one or both of the Hamiltonian and the quantum state includes applying a unitary transformation to said one or both of the Hamiltonian and the quantum state. 
     
     
         7 . The quantum optimization method of  claim 1 , further comprising generating, on the classical computer, the expectation value of each of the observables. 
     
     
         8 . The quantum optimization method of  claim 7 , further comprising updating, on the classical computer, a first representation of the quantum state based on the expectation value of the Hamiltonian to better approximate a ground state of the Hamiltonian. 
     
     
         9 . The quantum optimization method of  claim 8 , further comprising repeating:
 said generating the expectation value of each of the observables;   said transforming one or both of the Hamiltonian and the quantum state; and   said updating the first representation of the quantum state;   until the first representation of the quantum state has converged.   
     
     
         10 . The quantum optimization method of  claim 1 , wherein:
 the linear combination of the observables includes at least one observable with a zero weight that becomes non-zero due to said transforming the Hamiltonian; and   the expectation values of the observables include an expectation value for the at least one observable with a zero weight.   
     
     
         11 . The quantum optimization method of  claim 1 , wherein said transforming one or both of the Hamiltonian and the quantum state includes applying a fermionic transformation to said one or both of the Hamiltonian and the quantum state. 
     
     
         12 . The quantum optimization method of  claim 11 , the fermionic transformation including rotations of active orbitals. 
     
     
         13 . The quantum optimization method of  claim 11 , the fermionic transformation including transformations out of an active space to incorporate at least one of a core orbital and a virtual orbital. 
     
     
         14 . The quantum optimization method of  claim 11 , the fermionic transformation including rotations that respect one or more of an open-shell spin symmetry, a closed-shell spin symmetry, and a geometric symmetry. 
     
     
         15 . The quantum optimization method of  claim 11 , further comprising implementing a quantum subspace expansion technique. 
     
     
         16 . The quantum optimization method of  claim 11 , further comprising implementing a marginal projection technique. 
     
     
         17 . The quantum optimization method of  claim 11 , further comprising obtaining any of the expectation values the observables via orbital frames. 
     
     
         18 . The quantum optimization method of  claim 1 , wherein said transforming one or both of the Hamiltonian and the quantum state includes applying a Majorana fermionic transformation to said one or both of the Hamiltonian and the quantum state. 
     
     
         19 . The quantum optimization method of  claim 18 , further comprising minimizing the expectation value of the Hamiltonian using a Givens parameterization. 
     
     
         20 . The quantum optimization method of  claim 18 , further comprising minimizing the expectation value of the Hamiltonian using semidefinite programming. 
     
     
         21 . The quantum optimization method of  claim 1 , wherein said transforming one or both of the Hamiltonian and the quantum state includes applying a spin transformation to said one or both of the Hamiltonian and the quantum state. 
     
     
         22 . The quantum optimization method of  claim 1 , wherein the Hamiltonian is an Ising Hamiltonian configured for solving a combinatorial optimization problem. 
     
     
         23 . The quantum optimization method of  claim 1 , wherein said transforming one or both of the Hamiltonian and the quantum state includes minimizing the expectation value of the Hamiltonian estimated for the quantum state. 
     
     
         24 . The quantum optimization method of  claim 23 , wherein said minimizing the expectation value of the Hamiltonian includes minimizing the expectation value of the Hamiltonian using semidefinite programming. 
     
     
         25 . A computing system configured for quantum optimization, comprising:
 a processor;   a memory communicably coupled with the processor and storing machine-readable instructions that, when executed by the processor, control the computing system to:   estimate, for a quantum state, an expectation value of a Hamiltonian, expressible as a linear combination of observables, based on expectation values of the observables, and   transform one or both of the Hamiltonian and the quantum state to reduce the expectation value of the Hamiltonian estimated for the quantum state.   
     
     
         26 . The computing system of  claim 25 , further comprising a quantum computer that is communicably coupled with the processor and configured to measure the expectation value of each of the observables.