Systems and methods for preparation of normal distributions on quantum computers
A quantum computer program may receive a mean, a standard deviation, and a discretization for a target normal distribution; determine a number of iterations t based on the mean and standard deviation; identify a value n from the discretization and value X 0 from the mean; initialize n qubits in register q and an ancilla; apply a quantum Fourier transform and a +X 0 gate to q; apply a Y-Rotation with angle π/2 to the ancilla; apply a controlled +1 gate to q controlled by the ancilla; apply a Hadamard gate to the ancilla; measure the ancilla; set a value C to a value of the measured ancilla; repeat from the Y-Rotation t times; apply an inverse quantum Fourier transform to q; and output amplitudes of basis states of q as the normal distribution. The program may use qubit-scaling to reduce the number of iterations needed.
1. A method for preparation of normal distributions on a quantum computer, comprising:
receiving, at a quantum computer program, a mean, a standard deviation, and a discretization;
determining, by the quantum computer program, a number of iterations based on the mean and the standard deviation;
identifying, by the quantum computer program, a first value from the discretization;
setting, by the quantum computer program, a plurality of qubits in a register equal to the first value;
identifying, by the quantum computer program, a value X 0 from the mean;
initializing, by the quantum computer program, a state of the plurality of qubits in the register and a state of an ancilla qubit to |0>;
applying, by the quantum computer program, a first Hadamard gate to the plurality of qubits in the register;
applying, by the quantum computer program, a quantum Fourier transform to the register;
applying, by the quantum computer program, a +X 0 gate to the register;
applying, by the quantum computer program, a Y-Rotation operation with an angle π/2 to the ancilla qubit;
applying, by the quantum computer program, a controlled +1 gate to the register controlled by the ancilla qubit;
applying, by the quantum computer program, a second Hadamard gate to the ancilla qubit;
measuring, by the quantum computer program, a value of the ancilla qubit;
applying, by the quantum computer program, an inverse quantum Fourier transform to the register; and
outputting, by the quantum computer program, a normal distribution, wherein the normal distribution comprises amplitudes of basis states of the plurality of qubits in the register.
2. The method of claim 1 , wherein the state of the register is
cos θ| x>| 0>+sin θ| x>| 1>
where |x> is the state of the register after the Y-Rotation operation.
3. The method of claim 1 , wherein the state of the register and the ancilla qubit is:
cos θ| x>| 0>+sin θ| x+ 1>|1>
after the application of the controlled +1 gate.
4. The method of claim 1 , wherein the state of the register and the ancilla qubit is:
1
2
(
cos
θ
❘
"\[LeftBracketingBar]"
x
〉
❘
"\[RightBracketingBar]"
0
〉
+
sin
θ
❘
"\[LeftBracketingBar]"
x
+
1
〉
❘
"\[RightBracketingBar]"
0
〉
)
+
1
2
(
cos
θ
❘
"\[LeftBracketingBar]"
x
〉
❘
"\[RightBracketingBar]"
1
〉
-
sin
θ
❘
"\[LeftBracketingBar]"
x
+
1
〉
❘
"\[RightBracketingBar]"
1
〉
)
after the application of the first Hadamard gate.
5. The method of claim 1 , wherein the steps of applying the Y-Rotation operation with the angle π/2 to the ancilla qubit, applying the controlled +1 gate to the register controlled by the ancilla qubit, applying the second Hadamard gate to the ancilla qubit, and measuring the ancilla qubit are repeated for the number of iterations.
6. The method of claim 1 , wherein the plurality of qubits in the register are post-selected based on the measured values of the ancilla qubit.
7. A system comprising:
a quantum computer; and
a classical computer comprising a processor and a memory, wherein the classical computer causes the quantum computer to:
receive a mean, a standard deviation, and a discretization;
determine a number of iterations based on the mean and the standard deviation;
identify a first value from the discretization;
set a plurality of qubits in a register equal to the first value;
identify a value X 0 from the mean;
initialize a state of the plurality of qubits in the register and a state of an ancilla qubit to |0>;
apply a first Hadamard gate to the plurality of qubits in the register;
apply a quantum Fourier transform to the register;
apply a +X 0 gate to the register;
apply a Y-Rotation operation with an angle π/2 to the ancilla qubit;
apply a controlled +1 gate to the register controlled by the ancilla qubit;
apply a second Hadamard gate to the ancilla qubit;
measure a value of the ancilla qubit;
apply an inverse quantum Fourier transform to the register; and
output a normal distribution, wherein the normal distribution comprises amplitudes of basis states of the plurality of qubits in the register.
8. The system of claim 7 , wherein the state of the register is
cos θ| x>| 0>+sin θ| x>| 1>
where |x> is the state of the register after the Y-Rotation operation.
9. The system of claim 7 , wherein the state of the register and the ancilla qubit is
cos θ| x>| 0>+sin θ| x+ 1>|1>
after the application of the controlled +1 gate.
10. The system of claim 7 , wherein the state of the register and the ancilla qubit is:
1
2
(
cos
θ
❘
"\[LeftBracketingBar]"
x
〉
❘
"\[RightBracketingBar]"
0
〉
+
sin
θ
❘
"\[LeftBracketingBar]"
x
+
1
〉
❘
"\[RightBracketingBar]"
0
〉
)
+
1
2
(
cos
θ
❘
"\[LeftBracketingBar]"
x
〉
❘
"\[RightBracketingBar]"
1
〉
-
sin
θ
❘
"\[LeftBracketingBar]"
x
+
1
〉
❘
"\[RightBracketingBar]"
1
〉
)
after the application of the first Hadamard gate.
11. The system of claim 7 , wherein applying the Y-Rotation operation with the angle π/2 to the ancilla qubit, applying the controlled +1 gate to the register controlled by the ancilla qubit, applying the second Hadamard gate to the ancilla qubit, and measuring the ancilla qubit are repeated for the number of iterations.
12. The system of claim 7 , wherein the plurality of qubits in the register are post-selected based on the measured values of the ancilla qubit.