Quantum Phase Estimation Without Control Qubits
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Solution Overview
Problem
Existing quantum computing methods for phase estimation, such as phase kickback, incur high costs in terms of gate operations and qubit connectivity, especially in Noisy, Intermediate-Scale Quantum (NISQ) devices, due to the need for control qubits and complex circuit arrangements.
Innovation Solution
The introduction of Algebraic Phase Kickback (APK), which utilizes algebraic relationships between unitary operators E and S to perform phase estimation without requiring control qubits or prior knowledge of eigenstates, by preparing a resource state that encodes phase information using a specific symmetry of the unitary matrix.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If traditional phase kickback methods are used with control qubits, then phase estimation can be performed, but circuit complexity and gate overhead increase significantly
Solution Approach 1:
The patent extracts and removes the control qubit from the phase estimation circuit, eliminating the need for controlled-unitary operations. Instead of using a control qubit to accumulate phase information through repeated applications, the method directly measures phase kickback from a single application of the unitary operator on an eigenstate, thereby simplifying the circuit while maintaining phase estimation capability
Solution Approach 2:
The patent introduces an intermediary measurement process where the phase information is extracted through measuring the expectation value of the unitary operator on the eigenstate. This intermediary measurement approach replaces the direct control qubit mechanism, allowing phase information to be obtained without the complexity of controlled operations and multiple qubit interactions
2Loss of information
If control qubits are used for phase kickback, then phase information can be accumulated, but qubit connectivity requirements and gate costs increase
Solution Approach 1:
The control qubit mechanism is extracted and removed from the system. Phase information accumulation is achieved not through repeated controlled applications that require high gate overhead, but through a single application followed by measurement of the expectation value, dramatically reducing the number of gates required
Solution Approach 2:
The eigenstate itself serves the dual purpose of being both the input state and the carrier of phase information. By measuring the expectation value of the unitary operator on this eigenstate, the system extracts phase information directly from the state's own properties without requiring external control qubits or repeated operations, reducing energetic cost
3Measurement precision
If multiple qubits are used in the control register, then phase estimation resolution improves, but the number of qubits and circuit resources scale unfavorably
Solution Approach 1:
The entire control register is extracted and removed from the circuit. Instead of using multiple qubits to encode phase information with high resolution, the method achieves phase estimation by measuring the expectation value directly, which inherently provides phase information without requiring multiple qubits for encoding
Solution Approach 2:
The mechanical quantum circuit system with multiple qubits and controlled operations is replaced with a measurement-based approach. By measuring the expectation value of the unitary operator on the eigenstate, phase information is obtained through classical measurement data rather than through quantum mechanical interference of multiple qubits, reducing the quantum resource requirements
Data Source
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AI summary
Disclosed herein are methods and systems for improved phase estimation in quantum computing. For example, included herein are methods for determining phase information of a matrix representative of a quantum circuit. In such examples, the method comprises: receiving, a unitary matrix, UM representative of a quantum circuit, U which acts or a target set of qubits and identifying generalised Pauli operators, E and S associated with the unitary matrix, UM, the matrices satisfying a first set of conditions: ES = -SE; EUM = UM-1E;and SUM = UMS;. The method further comprises defining an input state |ψ〉 to the quantum circuit, U represented by the unitary matrix UM, wherein the input state |ψ〉 has support confined to the +1 or -1 eigenspace of S; obtaining phase information corresponding to the quantum circuit, U by: preparing a resource state |ψ〉r = |ψ〉 + eix E |ψ〉and applying UM to the resource state a predetermined number of times to provide an output state encoding eigenvalues eiϕi of the quantum circuit U.