The partial oracles framework is a quantum search algorithm that has the potential to exceed the quadratic speedup of Grover's algorithm, up to a theoretical maximum of an exponential speedup. Until now, however, the framework has lacked an explicit method for constructing the operator that represents the search iteration. In this paper, we provide the missing construction, for the special case of an oracle function definable using only in-place operations (that is, where the calculated result of the oracle function can be read just from the qubits in the search index). The restriction to in-place operations means that the current work does not yet exhibit quantum advantage: oracle functions constructed using only in-place operations are always classically reversible. To demonstrate quantum advantage, it will be necessary to extend this construction method to include out-of-place operations (part II). As part of the construction of the search iteration operator, we define a new type of transform, the reciprocal transform, which is applied to the oracle function. We show that the reciprocal transform obeys a chain rule, which makes it possible to break down complex transforms into simple steps. To illustrate the practical application of this search method, we apply the reciprocal transform to elementary operations from the SHA-256 hash algorithm: addition modulo $2^n$ , the $Maj(a, b, c)$ function, the $Ch(a, b, c)$ function, and the bit shift functions. We also introduce the QFrame python library, which is used to automate the construction of quantum circuits that represent reciprocal transforms.The partial oracles framework is a quantum search algorithm that has the potential to exceed the quadratic speedup of Grover's algorithm, up to a theoretical maximum of an exponential speedup. Until now, however, the framework has lacked an explicit method for constructing the operator that represents the search iteration. In this paper, we provide the missing construction, for the special case of an oracle function definable using only in-place operations (that is, where the calculated result of the oracle function can be read just from the qubits in the search index). The restriction to in-place operations means that the current work does not yet exhibit quantum advantage: oracle functions constructed using only in-place operations are always classically reversible. To demonstrate quantum advantage, it will be necessary to extend this construction method to include out-of-place operations (part II). As part of the construction of the search iteration operator, we define a new type of transform, the reciprocal transform, which is applied to the oracle function. We show that the reciprocal transform obeys a chain rule, which makes it possible to break down complex transforms into simple steps. To illustrate the practical application of this search method, we apply the reciprocal transform to elementary operations from the SHA-256 hash algorithm: addition modulo $2^n$, the $Maj(a, b, c)$ function, the $Ch(a, b, c)$ function, and the bit shift functions. We also introduce the QFrame python library, which is used to automate the construction of quantum circuits that represent reciprocal transforms.

quant-pharXiv:2604.21788Apr 24, 20260 citations

Quantum Instruction Simulation

Nadia Nedjah, Luiza de Macedo Mourelle

This chapter discusses the execution of a quantum operation that manipulates a single quantum bit, showing the time diagram during its simulation. It illustrates the execution of an operation on non-entangled quantum bits, showing the time diagram during its simulation. When performing a quantum operation on two or more q-bits where at least one of them is entangled with another and/or when the quantum operation causes entanglement, the construction of the quantum operator by means of a tensor product is required. The coefficients of the basic quantum operators are read from memory MOP and stored in memory MSC when the operator is constructed for two q-bits. The construction of an operator for three or more q-bits requires the reading and writing of the coefficients in MSC, eventually occurring simultaneously, but always at different addresses.This chapter discusses the execution of a quantum operation that manipulates a single quantum bit, showing the time diagram during its simulation. It illustrates the execution of an operation on non-entangled quantum bits, showing the time diagram during its simulation. When performing a quantum operation on two or more q-bits where at least one of them is entangled with another and/or when the quantum operation causes entanglement, the construction of the quantum operator by means of a tensor product is required. The coefficients of the basic quantum operators are read from memory MOP and stored in memory MSC when the operator is constructed for two q-bits. The construction of an operator for three or more q-bits requires the reading and writing of the coefficients in MSC, eventually occurring simultaneously, but always at different addresses.

CRC Press eBooksJan 5, 20230 citations

The Operators of Quantum Mechanics

Philip Franklin

Studies in Applied MathematicsApr 1, 19291 citation

Operators in Quantum Theory

J. L. B. Cooper

NatureJul 14, 19452 citations

Quantum Action-Angle Variables for the Harmonic Oscillator

Harry R. Lewis, W. E. Lawrence, Joseph David Harris

Operators conjugate to the Hamiltonian are constructed explicitly for the quantum harmonic oscillator by two approaches in the space spanned by the eigenstates of $q$ and the eigenstates of $p$ . The operators are quantum analogs of a classical angle variable divided by the oscillator frequency. Matrix elements have been evaluated in the coherent state representation. Either conjugate operator can be used to construct an explicitly time-dependent operator invariant. It can be used as the starting point in a new perturbative procedure for constructing invariant operators for nonlinear, nonautonomous Hamiltonians.Operators conjugate to the Hamiltonian are constructed explicitly for the quantum harmonic oscillator by two approaches in the space spanned by the eigenstates of $q$ and the eigenstates of $p$. The operators are quantum analogs of a classical angle variable divided by the oscillator frequency. Matrix elements have been evaluated in the coherent state representation. Either conjugate operator can be used to construct an explicitly time-dependent operator invariant. It can be used as the starting point in a new perturbative procedure for constructing invariant operators for nonlinear, nonautonomous Hamiltonians.

Physical Review LettersDec 23, 199627 citations

2^n

, the

Maj(a, b, c)

function, the

Ch(a, b, c)

function, and the bit shift functions. We also introduce the QFrame python library, which is used to automate the construction of quantum circuits that represent reciprocal transforms.

Operators conjugate to the Hamiltonian are constructed explicitly for the quantum harmonic oscillator by two approaches in the space spanned by the eigenstates of

q

and the eigenstates of

p

. The operators are quantum analogs of a classical angle variable divided by the oscillator frequency. Matrix elements have been evaluated in the coherent state representation. Either conjugate operator can be used to construct an explicitly time-dependent operator invariant. It can be used as the starting point in a new perturbative procedure for constructing invariant operators for nonlinear, nonautonomous Hamiltonians.