Quantum algorithm utilizing lattice Boltzmann method

1 . A method for setting up a quantum circuit utilizing a lattice Boltzmann method, the method comprising:

performing an initialization step by setting up for the quantum circuit, a first quantum register (f1), a second quantum register (f2) and at least a first ancilla register (a, a1), wherein the first quantum register (f1) comprises four qubits (f1 0 , f1 1 , f1 2 , f1 3 ), the second quantum register (f2) comprises N−4 qubits (f2 0 , . . . f2 N-5 ), and the first ancilla register (a, a1) comprises an ancilla qubit; and

applying the quantum circuit to the first quantum register (f1), the second quantum register (f2) and the first ancilla register (a, a1), wherein the quantum circuit comprises in a sequential order:

(i) a collision step constructed using a first set of quantum gates;

(ii) a propagation step constructed using a second set of quantum gates; and

(iii) a macroscopic variables calculation step constructed using a third set of quantum gates; and

wherein the first set of quantum gates comprises in a sequential order:

a first SWAP gate that is applied between a last working qubit (f2 N-5 ) of the second quantum register (f2) and the ancilla qubit of the first ancilla register (a, a1);

a first rotation operator (Ry) gate that has a Y-axis as a rotation axis, and that uses the ancilla qubit of the first ancilla register (a, a1) as a control corresponding to a state of |0> and uses the last working qubit (f2 N-5 ) of the second quantum register (f2) as a target;

a second rotation operator (Ry) gate that has the Y-axis as a rotation axis, and that uses the ancilla qubit of the first ancilla register (a, a1) as a control corresponding to a state of |1> and uses the last working qubit (f2 N-5 ) of the second quantum register (f2) as a target;

a first CX gate that uses the ancilla qubit of the first ancilla register (a, a1) as a control corresponding to a state of |1>, and the last working qubit (f2 N-5 ) of the second quantum register (f2) as a target; and

a second SWAP gate that is applied between the last working qubit (f2 N-5 ) of the second quantum register (f2) and the ancilla qubit of the first ancilla register (a, a1).

2 . The method according to claim 1 , further comprising simulating a physical process by performing the collision step, the propagation step and the macroscopic variables calculation step of the quantum circuit.

3 . The method according to claim 2 , further comprising:

generating a mathematical formulation corresponding to the physical process;

numerically discretizing the mathematical formulation by using the lattice Boltzmann method; and

constructing the collision step, the propagation step and the macroscopic variables calculation step of the quantum circuit, based on the numerically-discretized mathematical formulation.

4 . The method according to claim 3 , wherein the mathematical formulation is a one-dimensional advection-diffusion equation.

5 . The method according to claim 1 , wherein the propagation step implements a computational basis state shift.

6 . The method according to claim 1 , wherein the third set of quantum gates comprises in a sequential order:

a SWAP gate that is applied between the last working qubit (f2 N-5 ) of the second quantum register (f2) and the ancilla qubit of the first ancilla register (a, a1); and

a Hadamard gate that is applied to the ancilla qubit of the first ancilla register (a, a1).

7 . The method according to claim 1 , wherein the second set of quantum gates comprises in a sequential order:

an initial step comprising four CX gates and a first X gate;

a first segment comprising seven multi-controlled gates;

if N>5, a second segment comprising a first set of CX gates, a second set of CX gates, and a cascade of gates arranged between the first set and the second set; and

a third segment comprising two CX gates and a second X gate.

8 . The method according to claim 7 , wherein the four CX gates of the initial step are applied in a sequential order of:

a first CX gate of the initial step that uses an ancilla qubit of the first ancilla register (a1) as a control corresponding to a state of |1>, and an ancilla qubit of a second ancilla register (a2) as a target;

a second CX gate of the initial step that uses a first working qubit (f1 0 ) of the first quantum register (f1) as a control corresponding to a state of |1>, and the ancilla qubit of the first ancilla register (a1) as a target;

a third CX gate of the initial step that uses a last working qubit (f2 N-5 ) of the second quantum register (f2) as a control corresponding to a state of |1>, and the ancilla qubit of the first ancilla register (a1) as a target; and

a fourth CX gate of the initial step that uses the ancilla qubit of the first ancilla register (a1) as a control corresponding to a state of |1>, and a second working qubit (f1 1 ) of the first quantum register (f1) as a target; and

the first X gate of the initial step is applied to the first working qubit (f1 0 ) of the first quantum register (f1).

9 . The method according to claim 7 , wherein the seven multi-controlled gates of the first segment are applied in a sequential order of: a first multi-controlled gate, a second multi-controlled gate, a third multi-controlled gate, a fourth multi-controlled gate, a fifth multi-controlled gate, a sixth multi-controlled gate, and a seventh multi-controlled gate, wherein:

the first multi-controlled gate and the fifth multi-controlled gate use the last working qubit (f2 N-5 ) of the second quantum register (f2) as a first control corresponding to a state of |0>, the ancilla qubit of the first ancilla register (a1) as a second control corresponding to a state of |1>, and the second working qubit (f1 1 ) of the first quantum register (f1) as a target;

the second multi-controlled gate and the fourth multi-controlled gate use the last working qubit (f2 N-5 ) of the second quantum register (f2) and the ancilla qubit of the first ancilla register (a1) as controls corresponding to a state of |1>, and a third working qubit (f1 2 ) of the first quantum register (f1) as a target;

the third multi-controlled gate uses the second working qubit (f1 1 ) of the first quantum register (f1), the third working qubit (f1 2 ) of the first quantum register (f1) and the ancilla qubit of the first ancilla register (a1) as controls corresponding to a state of |1>, and a fourth working qubit (f1 3 ) of the first quantum register (f1) as a target;

the sixth multi-controlled gate uses the second working qubit (f1 1 ) of the first quantum register (f1) and the last working qubit (f2 N-5 ) of the second quantum register (f2) as controls corresponding to a state of |0>, the ancilla qubit of the first ancilla register (a1) as a control corresponding to a state of |1>, and the third working qubit (f1 2 ) of the first quantum register (f1) as a target; and

the seventh multi-controlled gate uses the second working qubit (f1 1 ) of the first quantum register (f1), the last working qubit (f2 N-5 ) of the second quantum register (f2) and the ancilla qubit of the first ancilla register (a1) as controls corresponding to a state of |1>, and the third working qubit (f1 2 ) of the first quantum register (f1) as a target.

10 . The method according to claim 7 , wherein the first set of CX gates of the second segment comprises four CX gates that use in following order: the first working qubit (f1 0 ), the second working qubit (f1 1 ), the third working qubit (f1 2 ) and the fourth working qubit (f1 3 ) of the first quantum register (f1) as respective targets, each of the four CX gates of the first set of CX gates using the last working qubit (f2 N-5 ) of the second quantum register (f2) as a control corresponding to a state of |0>;

wherein the second set of CX gates of the second segment comprises N−2 CX gates that use in following order: the first working qubit (f1 0 ), the second working qubit (f1 1 ), the third working qubit (f1 2 ) and up to (N−2) th working qubit as respective targets, each of the N−2 CX gates of the second set of CX gates using the last working qubit (f2 N-5 ) of the second quantum register (f2) as a control corresponding to a state of |0>; and

wherein the cascade of gates is arranged after the first set of CX gates and before the second set of CX gates, and wherein the number of gates in the cascade is a function of a total number (N) of working qubits.

11 . The method according to claim 10 , wherein the cascade of the gates comprises:

N−5 multi-controlled gates arranged as a sequence of gates indexed from 1 to N−5, wherein within the sequence, a multi-controlled gate indexed ‘M’ uses as controls all working qubits in the first quantum register (f1) and M−1 first working qubits of the second quantum register, wherein the controls correspond to a state of |1>, and uses as a target M th working qubit of the second quantum register (f2); and

N−6 CX gates, if N is greater than 6,

wherein the N−5 multi-controlled gates and the N−6 CX gates are arranged in an interleaved manner, the cascade of the gates starting with a first multi-controlled gate from amongst the N−5 multi-controlled gates, each CX gate using the last working qubit (f2 N-5 ) of the second quantum register (f2) as a control corresponding to a state of |0>, wherein an L th CX gate uses as a target a same working qubit that is used as a target by an L th multi-controlled gate.

12 . The method according to claim 7 , wherein the two CX gates and the second X gate of the third segment are applied in a sequential order of:

a first CX gate of the third segment that uses the last working qubit (f2 N-5 ) of the second quantum register (f2) as a control corresponding to a state of |1>, and the ancilla qubit of the first ancilla register (a1) as a target; and

a second CX gate of the third segment that uses the first working qubit (f1 0 ) of the first quantum register (f1) as a control corresponding to a state of |1>, and the ancilla qubit of the first ancilla register (a1) as a target,

wherein the second X gate of the third segment is applied to the ancilla qubit of the first ancilla register (a1).

13 . A quantum computer or a quantum simulator configured to execute a method according to claim 1 .

14 . A non-transitory computer-readable medium having computer program instructions stored thereon, the computer program instructions being executable by at least one processor in a classical computer to control a quantum computer or a quantum simulator to perform a method according to claim 1 .