Quantum phase estimate is a quantum method for estimating the eigenvalue of a unitary operator. Many quantum algorithms depend on this fundamental subroutine, which uses the QFT to convert phase information into an estimate with the precision of the estimate contingent on the qubit count. The phase of an eigenvector may be found and quantum system dynamics can be simulated using the method.
Concepts of Quantum Phase Estimation
Unitary Operator: A unitary operator, sometimes abbreviated as U, is a transformation preserving quantum state length. In quantum physics, a closed system develops in unitary fashion.
Eigenvector and Eigenvalue: An eigenvector of U is a quantum state scaled by a complex integer, hence defining the eigenvalue of U when operated upon. U|ψ⟩ = λ|ψ⟩ if |ψ⟩ is an eigenvector of U having an eigenvalue λ. U is unitary, hence the eigenvalue λ has a magnitude of 1 and may be expressed as λ = e(2πiφ), in which case φ is a real integer between 0 and 1, known as the phase.
Quantum Fourier Transform (QFT): One fundamental component of QPE is the Quantum Fourier Transform (QFT). Operating on quantum amplitudes rather than classical signals, it is the quantum equivalent of the conventional Discrete Fourier Transform and a fundamental instrument in quantum algorithms. Comparatively to conventional methods that usually demand O(n2n), the QFT is rather efficient and scales as O(n2) for n qubits. Phase information of the quantum system is encoded and manipulated using the QFT. The last phase extraction stage of QPE uses its inverse operation, the Inverse Quantum Fourier Transform (IQFT),.
Relation to Quantum Fourier Transform (QFT): Phase estimate makes most use of the QFT. The phase information stored is turned into an estimate of the phase via the QFT by means of quantum state transformation. Extensive phase information is obtained via the inverse QFT (IQFT). In quantum computing, the QFT is an indispensable instrument. Key for many quantum algorithms, polyn time allows one to do it.
Phase Kickback: Phase kickback is the phenomena when a controlled-U operation performed to an eigenstate of U results in One can transmit the phase connected with the eigenvalue of U to the control qubit.
How does work quantum phase estimation in quantum computing
The quantum phase estimate method approximates the phase φ of a unitary operator U. Input for the procedure is a unitary operator U and one of its eigenvectors |ψ⟩ such that U|ψ⟩ = e(2πiφ) |ψ⟩, where φ is the phase to be calculated. One wants to roughly estimate φ with a desired accuracy.
The quantum phase estimating algorithm follows steps:
Initialization: Get two quantum registers first. First register consists of n qubits initially set to |0⟩; second register carries the eigenvector |ψ⟩ of the unitary operator U. The precision of the phase estimation is found in the first register’s qubit count n.
Superposition: Using a Hadamard gate on every qubit in the first register generates a superposition state. This generates the state (1/√(2n))∑|x⟩ where x runs from 0 to 2n-1.
Controlled Unitary Operations: Apply a sequence of controlled-U operations whereby U acts on the second register and the control qubits come from the first register. The unitary U^(x) is applied to the second register for a control qubit in state |x⟩. Every qubit in the first register regulates an other power of U. Particularly, the k-th qubit (counting from 0) regulates U(2k). The state of the registers is
(1/√(2n))∑e(2πiφx)|x⟩|ψ⟩
Inverse Quantum Fourier Transform (IQFT): Apply an inverse quantum Fourier transform on the first register in 4. The IQFT translates the phase information contained in the superposition to an estimate of the phase, therefore acting in reverse of the QFT.
Measurement: Get the first register in the computational basis measured. The output offers a phasing φ estimate. Should φ be precisely specified with n bits of accuracy, the measurement will provide the binary expansion of φ. The method will provide an approximation of the phase with great probability if the phase cannot be precisely described using n bits of accuracy.
Quantum Phase Estimation Without Controlled Unitaries
Input: The algorithm takes as input a unitary operator U and one of its eigenvectors |ψ⟩ such that
U|ψ⟩ = e2πiφ|ψ⟩,
where φ is the phase to be estimated.
Initialization: QPE begins by preparing two quantum registers. The first register, which consists of n qubits, is initialized to the state |0⟩n. The second register is initialized to the state |ψ⟩, the eigenvector of U. The number of qubits n in the first register determines the precision of the phase estimate.
Superposition: A Hadamard gate is applied to each qubit in the first register, creating a superposition state7. This is equivalent to applying the QFT to the all-zero state in the first register. This results in the state (1/√2n) ∑|x⟩|ψ⟩, where x ranges from 0 to 2n-1.
Phase Encoding: The unitary operator U is applied to the second register. This results in the state (1/√2n)∑e2πiφx|x⟩|ψ⟩ where the phase information is now encoded in the amplitudes of the superposition.
Inverse Quantum Fourier Transform: The inverse quantum Fourier transform (IQFT) is applied to the first register. The IQFT is the reverse operation of the QFT and is used to extract the phase information encoded in the amplitudes of the superposition, mapping it to an estimate of the phase.
Measurement: The first register is measured in the computational basis. The measurement outcome will be an estimate of φ, where the precision of the estimate is determined by n, the number of qubits in the first register
Mathematical Representation
Let eigenvalue λ = e(2πiφ) from the eigenvector |ψ⟩ of a unitary operator U.
- The algorithm prepares the state |0⟩n|ψ⟩.
- Following Hadamard gates to the first register, the state becomes (1/√(2n)).∑|x⟩ where x spans 0 to 2n-1.
- Then controlled-U operations are used producing the state (1/√(2n))∑e(2πiφx)|x⟩|ψ⟩.
- First register is treated with the IQFT. Should φ be expressed with n bits, the measurement of the first register generates an estimate φ̃ such that φ = φ̃.
•The binary expansion of φ/2π will be the outcome if its finishes after at most n bits. Should not be the case, an approximation will follow.
Phase Estimation Precision The precision of the phase estimation depends on the qubit count in the first register. The method returns the exact phase if the phase may be precisely expressed with n bits. The method will offer an approximation with great probability, nonetheless, if the phase calls for extra bits for correct representation. Even if the phase cannot be precisely expressed with n bits of accuracy, there is a great chance of reaching a good approximation.
Quantum Phase Estimation(QPE) uses
- Shor’s Algorithm: A quantum method for factoring big numbers, Shor’s algorithm depends much on phase estimate. Crucially, a stage in the factoring technique, it determines the order of an element in a multiplicative group.
- Eigenvalue Estimation: Phase estimate allows one to estimate the eigenvalues of unitary operators. Many quantum algorithms and quantum simulations depend on the efficient estimate of eigenvalues.
- Hamiltonian Simulation: Particularly in the energy of a quantum system, phase estimate may also be applied in quantum simulation. The method may implement the time-evolution operator and derive the eigenenergies considering a Hamiltonian.
- Amplitude Estimation: Phase estimation is a subroutine in amplitude estimation, a quantum method for approximating the amplitude of a quantum state.
- Quantum Counting: Quantum counting—the method used to count the solutions in a search space—can be accomplished with the amplitude estimate technique.
- Quantum Walks: Phase estimation finds use in quantum walk algorithms, quantum equivalents of classical random walks applied in search problem solutions.
- Quantum Machine Learning: Phase estimation can improve computer efficiency of models of machine learning.
- Variational quantum algorithms (VQAs) comprising variational quantum eigensolvers (VQEs) also make use of quantum phase estimation to assist in the ground state energy of molecule discovery. In this instance, the desired state is prepared by means of a quantum circuit using a trial wave function. A traditional optimization loop then modifies the quantum circuit’s parameters until the energy is lowest. Quantum phase estimation is also applied in quantum approximation optimization algorithms (QAOA), applied to handle combinatorial optimization issues.
- In materials science and quantum chemistry, the method finds value in several directions. Understanding the behavior of many-body systems depends on an estimation of the partition function, which may be obtained here. In quantum metrology, it may also be utilized to create very precise quantum measuring systems.
Practical Implementation: The implementation of quantum phase estimation is dependent on two factors, the accuracy of the estimate required and the probability of the algorithm succeeding. For practical applications, modular exponentiation can be used to implement the controlled-U^2x operations efficiently. The number of circuit repetitions may be optimized to minimize the Fisher information of the estimated parameter.
Limitations
- Phase estimating methods are prone to mistakes. When a phase cannot be stated with n bits of accuracy, errors in phase estimate result from the approximative character of the calculation.
- Decoherence: Decoherence could affect how the QFT performs in phase estimate. Decoherence describes the loss of quantum information brought on by environmental interaction.
- Input State: The procedure works properly only with a specified eigenstate of the unitary operator.