[Paper Review] Phase estimation as a quantum nondemolition measurement

B. C. Travaglione, G. J. Milburn+1 more|ArXiv.org|Mar 27, 2002

Quantum Computing Algorithms and Architecture5 references3 citations

TL;DR

This paper demonstrates that the quantum phase estimation algorithm—central to Shor's factoring and other quantum algorithms—is mathematically equivalent to a quantum nondemolition (QND) measurement. By showing that phase estimation satisfies all three criteria of QND measurements (commutation with free Hamiltonian, interaction Hamiltonian, and meter variable), the authors unify two key concepts in quantum information: quantum algorithm design and back-action-evading measurement, extending the equivalence to both discrete and continuous variable quantum systems.

ABSTRACT

The phase estimation algorithm, which is at the heart of a variety of quantum algorithms, including Shor's factoring algorithm, allows a quantum computer to accurately determine an eigenvalue of an unitary operator. Quantum nondemolition measurements are a quantum mechanical procedure, used to overcome the standard quantum limit when measuring an observable. We show that the phase estimation algorithm, in both the discrete and continuous variable setting, can be viewed as a quantum nondemolition measurement.

Motivation & Objective

• To establish a theoretical equivalence between the phase estimation algorithm and quantum nondemolition (QND) measurements.
• To demonstrate that phase estimation satisfies the three fundamental criteria of QND measurements: commutation with free Hamiltonian, interaction Hamiltonian, and non-commuting meter variable.
• To extend the equivalence to continuous variable quantum systems, showing that continuous variable phase estimation is isomorphic to continuous variable QND measurement.
• To unify concepts from quantum algorithm design and quantum measurement theory, highlighting shared principles in quantum information processing.

Proposed method

• Formalize the three criteria for a QND measurement: [ĤF, Â] = 0, [ĤSM, Â] = 0, and [ĤSM, B̂] ≠ 0.
• Apply the phase estimation algorithm in three stages: initialization (preparing meter in zero eigenstate), entanglement (applying controlled unitary Û = e^{iĤUΛ}), and measurement (Fourier transform and computational basis measurement).
• Map the discrete qubit phase estimation circuit to a QND measurement framework, using the Hadamard gate as the Fourier transform and controlled-NOT as the interaction Hamiltonian.
• Extend the analysis to continuous variables by replacing qubit registers with infinite-level systems, using position eigenstates as computational basis and momentum as conjugate meter variable.
• Show that the interaction Hamiltonian e^{iX̂IÂT} satisfies QND criteria, with  as the observable to be measured.
• Use the fact that the final measurement in the Fourier basis corresponds to measuring the conjugate variable, consistent with QND principles.

Experimental results

Research questions

• RQ1 Can the phase estimation algorithm be interpreted as a quantum nondemolition measurement?
• RQ2 Do the three defining criteria of QND measurements hold for the phase estimation protocol?
• RQ3 Is the equivalence between phase estimation and QND measurement preserved in the continuous variable regime?
• RQ4 How does the phase estimation algorithm's structure align with the back-action evasion principle in quantum measurement?
• RQ5 What is the operational and conceptual connection between quantum algorithms and precision measurement techniques?

Key findings

• The phase estimation algorithm satisfies all three criteria of a quantum nondemolition measurement: the observable commutes with the free Hamiltonian, the interaction Hamiltonian commutes with the observable, and the meter variable does not commute with the interaction Hamiltonian.
• In the discrete case, the controlled-NOT gate implements the interaction Hamiltonian e^{iĤSM} = e^{iχX̂aŶb}, which satisfies [ĤSM, X̂a] = 0 and [ĤSM, Ŷb] ≠ 0.
• The initial state |x⟩a|0⟩b evolves under the interaction to ∫dy |x⟩a|y⟩b, demonstrating entanglement between system and meter, consistent with QND measurement.
• The final measurement in the Fourier basis (via Hadamard gate) projects the meter variable onto the conjugate of the system observable, enabling eigenvalue extraction without back-action on the measured observable.
• In the continuous variable limit, the phase estimation protocol maps directly to an ideal quadrature QND measurement, with the index system prepared in a momentum eigenstate (zero momentum) and the final measurement in the position basis.
• The equivalence is approximate in practice due to the physical inaccessibility of ideal position eigenstates, requiring squeezed states or other approximations for experimental realization.