CLAIMS

1. A method performed by a classical computer for implementing, on a quantum computer, a non-unitary operation of the form / + aU, where / is the identity operator, a is a scalar, and U is a unitary operator, the quantum computer having a plurality of qubits, including an ancilla qubit,

the classical computer including a processor, a non-transitory computer-readable medium, and computer program instructions stored in the non-transitory computer-readable medium, the computer program instructions being executable by the processor to perform the method, the method comprising:

(A) generating and storing, in the non-transitory computer-readable medium, computer-readable data representing a description of a first quantum circuit W which, when executed by the quantum computer, probabilistically realizes the non-unitary operation by the technique of linear combination of unitaries;

(B) generating and storing, in the non-transitory computer-readable medium, computer-readable data representing a description of a second quantum circuit, the second quantum circuit comprising a sequence of quantum gates S_{d} =

parametrized by an integer d,

wherein

and wherein is

a single-qubit rotation applied to the ancilla qubit .

2. The method of claim 1, further comprising:

(C) on the quantum computer, executing the first quantum circuit to probabilistically realize the non-unitary operation by the technique of linear combination of unitaries.

3. The method of claim 2, further comprising:

(D) on the quantum computer, executing the second quantum circuit, comprising executing the sequence of quantum gates .

4. The method of claim 3, wherein (D) comprises, on the quantum computer:

(D) (1) applying a single-qubit rotation R_{a} to transform the ancilla qubit into the state

(D) (2) applying controlled-U operator on a subset of the plurality of qubits, not including the ancilla qubit, conditioned on the ancilla qubit being in the state |1); and

(D) (3) applying on the ancilla qubit.

5. The method of claim 1, further comprising generating and storing, in the non-transitory computer- readable medium, additional computer-readable data that, when executed on the quantum computer, causes the second quantum circuit to execute repeatedly, on the quantum computer, to perform a sequence of operations

, that approximates the operator with t being a

scalar and being the cluster operator which

is a linear combination of operators P_{j} parametrized by weights .

6. The method of claim 5, further comprising:

(C) performing mean- field approximation to generate and store, in the non-transitory computer- readable medium, computer-readable data representing a description of a third quantum circuit to prepare a reference state;

(D) generating and storing, in the non-transitory computer-readable medium, computer-readable data representing a description of a parametrized quantum circuit for approximating .

(E) on the quantum computer, executing the third quantum circuit to prepare the reference state;

(F) on the quantum computer, applying the

parametrized quantum circuit to the reference state to generate the ansatz state

;

(G) using the classical computer in cooperation with the quantum computer to measure an energy of the ansatz and

(H) on the classical computer, iteratively tuning the parameters to minimize the energy of the

ansatz

7. A system comprising:

a classical computer the classical computer

including a processor, a non-transitory computer-readable medium, and computer program instructions stored in the non-transitory computer-readable medium;

a quantum computer comprising a plurality of qubits, including an ancilla qubit;

wherein the computer program instructions, when executed by the processor, perform a method for

implementing, on the quantum computer, a non-unitary operation of the form / + aU , where / is the identity operator, a is a scalar, and U is a unitary operator, the method comprising:

(A) generating and storing, in the non-transitory computer-readable medium, computer-readable data representing a description of a first quantum circuit W which, when executed by the quantum computer, probabilistically realizes the non-unitary operation by the technique of linear combination of unitaries;

(B) generating and storing, in the non-transitory computer-readable medium, computer-readable data representing a description of a second quantum circuit, the second quantum circuit comprising a sequence of quantum gates

parametrized by an integer d,

wherein

and wherein is

a single-qubit rotation applied to the ancilla qubit .

8. The system of claim 7, wherein the method further comprises :

(C) on the quantum computer, executing the first quantum circuit to probabilistically realize the non-unitary operation by the technique of linear combination of unitaries.

9. The system of claim 8, wherein the method further comprises :

(D) on the quantum computer, executing the second quantum circuit, comprising executing the sequence of quantum gates

10. The system of claim 9, wherein (D) comprises, on the quantum computer:

(D) (1) applying a single-qubit rotation R_{a} to

transform the ancilla qubit into the state

(D) (2) applying controlled-U operator on a subset of the plurality of qubits, not including the ancilla qubit, conditioned on the ancilla qubit being in the state |1); and

(D) (3) applying on the ancilla qubit.

11. The system of claim 7, wherein the method further comprises generating and storing, in the non-transitory computer-readable medium, additional computer-readable data that, when executed on the quantum

computer, causes the second quantum circuit to execute repeatedly, on the quantum computer, to perform a sequence of operations that approximates

the operator with t being a scalar and

being the cluster operator which is a linear combination of operators P_{j} parametrized by weights

12. The system of claim 11, wherein the method further comprises:

(C) performing mean-field approximation to generate and store, in the non- transitory computer- readable medium, computer- readable data representing a description of a third quantum circuit to prepare a reference state; (D) generating and storing, in the non- transitory computer-readable medium, computer- readable data representing a description of a

parametrized quantum circuit for approximating

(E) on the quantum computer, executing the third quantum circuit to prepare the reference state;

(F) on the quantum computer, applying the

parametrized quantum circuit to the reference state to generate the ansatz state

(G) using the classical computer in cooperation with the quantum computer to measure an energy of the ansatz ; and

(H) on the classical computer, iteratively tuning the parameters to minimize the energy of the

ansatz .