Systems and methods for quantum computation of an extended weighted sum of hashed values of a real-valued function
Systems, apparatuses, and methods are disclosed for a quantum computation of an extended weighted sum of hashed function values. An example method includes initializing two sets of qubits. The example method also includes transforming the initial quantum states of the sets of qubits into a quantum state encoding the weighted sum of the hashed function values using a sequence of operators. The sequence of operators comprises a first operator that encodes a distribution of real-valued weights as quantum amplitudes in the quantum state of the first set of qubits, a second operator that prepares an entangled quantum state encoding a real-valued function, a third operator comprising a series of Hadamard gates, and a fourth operator whose conjugate encodes a real-valued hash function. The example method also includes utilizing the quantum state encoding the weighted sum of the hashed function values.
1 . A method for a quantum computation of a weighted sum of hashed values of a real-valued function ƒ, the method comprising:
initializing, via state initialization circuitry, a first set of n qubits and a second set of m qubits by preparing the first set of n qubits and the second set of m qubits in initial quantum states;
transforming, via state transformation circuitry, the initial quantum state of the first set of n qubits and the initial quantum state of the second set of m qubits into a quantum state encoding the weighted sum of hashed values of ƒ using a first sequence of operators comprising:
an operator A that encodes a distribution of real-valued weights;
an operator F′ that transforms the quantum state encoding the distribution of real-valued weights on the first set of n qubits and the initial quantum state of the second set of m qubits into an entangled quantum state encoding the real-valued function ƒ, wherein amplitudes of the entangled quantum state are real-valued, wherein the operator F′ comprises a one or more of an operator U γ (θ) that is controlled on a subset of qubits from the first set of n qubits, wherein the one or more of the operator U γ (θ) prepares a quantum state with amplitudes forming a geometric sequence representing the real-valued function ƒ,
an operator H n , comprising a series of Hadamard gates, and
an operator B † that encodes a real-valued hash function as quantum amplitudes; and
utilizing, via the state transformation circuitry, the quantum state encoding the weighted sum of hashed values of ƒ with a second sequence of operators, wherein the second sequence of operators alters or transfers the quantum state encoding the weighted sum of hashed values of ƒ.
2 . The method of claim 1 , wherein:
the operator A encodes the distribution of real-valued weights as quantum amplitudes a k in the quantum state of the first set of n qubits, and
the operator B † is a Hermitian conjugate of an operator B that encodes the real-valued hash function h as the quantum amplitudes b v , wherein applying the operators H n and B † to the first set of n qubits and second set of m qubits in the entangled quantum state results in the quantum state encoding the weighted sum of hashed values of ƒ.
3 . The method of claim 2 , wherein the operator F′ comprises:
an operator H n+m , wherein the operator H n+m , comprises a second series of Hadamard gates;
a third sequence of operators comprising the one or more of the operator U γ (θ),
an inverse quantum Fourier transform, wherein applying the inverse quantum Fourier transform to the second set of m qubits in the quantum state with quantum amplitudes forming the geometric sequence representing the real-valued function ƒ prepares a quantum state encoding the real-valued function ƒ in complex-valued amplitudes; and
an operator R l that, when applied to the first set of n qubits and the second set of m qubits in the quantum state encoding the real-valued function ƒ in complex-valued amplitudes, removes phase rotations to produce an entangled quantum state encoding the real-valued function ƒ with real-valued amplitudes.
4 . The method of claim 2 , wherein:
the weighted sum of hashed values of ƒ is an expected value of the real-valued function ƒ, and
the real-valued hash function h is an identity function.
5 . The method of claim 4 , wherein:
the weighted sum of hashed values of ƒ is a mean value of the real-valued function ƒ
values of the real-valued weights in the distribution of real-valued weights are equal to 1, and
the operator A comprises a second series of Hadamard gates.
6 . The method of claim 2 , wherein the weighted sum of hashed values of ƒ is a price of a financial product,
the real-valued function ƒ is a payoff function of a financial product, and
the real-valued hash function h returns zero for input values below a specified cutoff, and the real-valued hash function h returns a linear function for values at or above the specified cutoff.
7 . The method of claim 1 , wherein the second sequence of operators comprises:
an operator measuring the quantum state encoding the weighted sum of hashed values of ƒ;
an operator storing information from the quantum state encoding the extended weighted sum of hashed function values;
an operator further transforming the quantum state encoding the extended weighted sum of hashed function values with a quantum circuit; or
a combination thereof.
8 . The method of claim 1 , further comprising:
transpiling, via transpilation circuitry, the first sequence of operators to obtain a transpiled quantum circuit,
wherein the transpiled quantum circuit causes the transforming of the initial quantum state of the first set of n qubits and the initial quantum state of the second set of m qubits.
9 . The method of claim 1 , further comprising:
transpiling, via transpilation circuitry, the second sequence of operators to obtain a transpiled quantum circuit,
wherein the transpiled quantum circuit causes the utilizing of the quantum state encoding the weighted sum of hashed values of ƒ.
10 . The method of claim 1 , further comprising:
transpiling, via transpilation circuitry, the first sequence of operators and the second sequence of operators to obtain a transpiled quantum circuit,
wherein the transpiled quantum circuit causes the transforming of the initial quantum state of the first set of n qubits and the initial quantum state of the second set of m qubits ƒ.
11 . The method of claim 1 , further comprising:
acquiring, via communications hardware, user input providing attributes of:
the distribution of real-valued weights,
the real-valued function ƒ,
the real-valued hash function h, or
a combination thereof;
wherein transforming the initial quantum state of the first set of n qubits and the initial quantum state of the second set of m qubits utilizes the attributes acquired from the user input.
12 . An apparatus for a quantum computation of a weighted sum of hashed values of a real-valued function ƒ, the apparatus comprising:
state initialization circuitry configured to:
initialize a first set of n qubits and a second set of m qubits by preparing the first set of n qubits and the second set of m qubits in initial quantum states; and
state transformation circuitry configured to:
transform the initial quantum state of the first set of n qubits and the initial quantum state of the second set of m qubits into a quantum state encoding the weighted sum of hashed values of ƒ using a first sequence of operators comprising:
an operator A that encodes a distribution of real-valued weights;
an operator F′ that transforms the quantum state encoding the distribution of real-valued weights on the first set of n qubits and the initial quantum state of the second set of m qubits into an entangled quantum state encoding the real-valued function ƒ, wherein amplitudes of the entangled quantum state are real-valued, wherein the operator F′ comprises a one or more of an operator U γ (θ) that is controlled on a subset of qubits from the first set of n qubits, wherein the one or more of the operator U γ (θ) prepares a quantum state with amplitudes forming a geometric sequence representing the real-valued function ƒ,
an operator H n , comprising a series of Hadamard gates, and
an operator B † that encodes a real-valued hash function as quantum amplitudes, and
utilize the quantum state encoding the weighted sum of hashed values of ƒ with a second sequence of operators, wherein the second sequence of operators alters or transfers the quantum state encoding the weighted sum of hashed values of ƒ.
13 . The apparatus of claim 12 , wherein:
the operator A encodes the distribution of real-valued weights as quantum amplitudes a k in the quantum state of the first set of n qubits, and
the operator B † is a Hermitian conjugate of an operator B that encodes the real-valued hash function h as the quantum amplitudes b v , wherein applying the operators H n and B † to the first set of n qubits and second set of m qubits in the entangled quantum state results in the quantum state encoding the weighted sum of hashed values of ƒ.
14 . The apparatus of claim 13 , wherein the operator F′ comprises:
an operator H n+m , wherein the operator H n+m , comprises a second series of Hadamard gates;
a third sequence of operators comprising the one or more of the operator U γ (θ),
an inverse quantum Fourier transform, wherein applying the inverse quantum Fourier transform to the second set of m qubits in the quantum state with quantum amplitudes forming the geometric sequence representing the real-valued function ƒ prepares a quantum state encoding the real-valued function ƒ in complex-valued amplitudes; and
an operator R l , wherein the operator R l , when applied to the first set of n qubits and the second set of m qubits in the quantum state encoding the real-valued function ƒ in complex-valued amplitudes, removes phase rotations to produce an entangled quantum state encoding the real-valued function ƒ with real-valued amplitudes.
15 . The apparatus of claim 13 , wherein:
the weighted sum of hashed values of ƒ is an expected value of the real-valued function ƒ, and
the real-valued hash function h is an identity function.
16 . The apparatus of claim 15 , wherein:
the weighted sum of hashed values of ƒ is a price of a financial product,
the real-valued function ƒ is a payoff function of a financial product, and
the real-valued hash function h returns zero for input values below a specified cutoff, and the real-valued hash function h returns a linear function for values at or above the specified cutoff.
17 . The apparatus of claim 13 , wherein the state transformation circuitry is further configured such that the second sequence of operators comprises:
operators measuring the quantum state encoding the weighted sum of hashed values of ƒ;
operators storing information from the quantum state encoding the extended weighted sum of hashed function values;
operators further transforming the quantum state encoding the extended weighted sum of hashed function values with a quantum circuit; or
a combination thereof.
18 . The apparatus of claim 12 , further comprising:
transpilation circuitry configured to:
transpile the first sequence of operators and the second sequence of operators to obtain a transpiled quantum circuit,
wherein the transpiled quantum circuit causes the transforming of the initial quantum state of the first set of n qubits and the initial quantum state of the second set of m qubits and the utilizing of the quantum state encoding the weighted sum of hashed values of ƒ.
19 . The apparatus of claim 12 , further comprising:
communications hardware, configured to:
acquire user input providing attributes of:
a distribution of real-valued weights,
the real-valued function ƒ,
a real-valued hash function h, or
a combination thereof;
wherein the state transformation circuitry is further configured such that transforming the initial quantum state of the first set of n qubits and the initial quantum state of the second set of m qubits utilizes the attributes acquired from the user input.
20 . An apparatus for a quantum computation of a weighted sum of hashed values of a real-valued function ƒ, the apparatus comprising:
means for initializing a first set of n qubits and a second set of m qubits by preparing the first set of n qubits and the second set of m qubits in initial quantum states;
means for transforming the initial quantum state of the first set of n qubits and the initial quantum state of the second set of m qubits into a quantum state encoding the weighted sum of hashed values of ƒ using a first sequence of operators comprising:
an operator A that encodes a distribution of real-valued weights;
an operator F′ that transforms the quantum state encoding the distribution of real-valued weights on the first set of n qubits and the initial quantum state of the second set of m qubits into an entangled quantum state encoding the real-valued function ƒ, wherein amplitudes of the entangled quantum state are real-valued, wherein the operator F′ comprises a one or more of an operator U γ (θ) that is controlled on a subset of qubits from the first set of n qubits, wherein the one or more of the operator U γ (θ) prepares a quantum state with amplitudes forming a geometric sequence representing the real-valued function ƒ,
an operator H n , comprising a series of Hadamard gates, and
an operator B † that encodes a real-valued hash function as quantum amplitudes; and
means for utilizing the quantum state encoding the weighted sum of hashed values of ƒ with a second sequence of operators, wherein the second sequence of operators alters or transfers the quantum state encoding the weighted sum of hashed values of ƒ.