[Paper Review] Natural parameterized quantum circuit

Tobias Haug, M. S. Kim|arXiv (Cornell University)|Jul 29, 2021

Quantum Computing Algorithms and Architecture2 references5 citations

TL;DR

This paper introduces the Natural Parameterized Quantum Circuit (NPQC), a hardware-efficient quantum circuit with a Euclidean quantum geometry at a reference parameter, enabling gradient descent to directly yield the quantum natural gradient (QNG) without computing the quantum Fisher information metric (QFIM). This leads to significantly faster initial training in variational quantum algorithms and enables efficient parameter estimation via circuit sampling, while also achieving the minimal quantum Cramér-Rao bound for multi-parameter metrology.

ABSTRACT

Noisy intermediate scale quantum computers are useful for various tasks such as state preparation and variational quantum algorithms. However, the non-Euclidean quantum geometry of parameterized quantum circuits is detrimental for these applications. Here, we introduce the natural parameterized quantum circuit (NPQC) that can be initialised with a Euclidean quantum geometry. The initial training of variational quantum algorithms is substantially sped up as the gradient is equivalent to the quantum natural gradient. Further, we show how to estimate the parameters of the NPQC by sampling the circuit, which could be used for benchmarking or calibrating NISQ hardware. For a general class of quantum circuits, the NPQC has the minimal quantum Cramér-Rao bound which highlights its potential for quantum metrology. Finally, we show how to generate arbitrary superpositions of two states with the NPQCs for state preparation tasks. Our results can be used to enhance currently available quantum processors.

Motivation & Objective

• To address the slow convergence of variational quantum algorithms (VQAs) due to non-Euclidean quantum geometry in parameterized quantum circuits (PQCs).
• To develop a parameterized quantum circuit with a Euclidean quantum Fisher information metric (QFIM) at a reference point, enabling natural gradient optimization without QFIM computation.
• To enable efficient, parallel parameter estimation from computational basis sampling for NISQ device calibration.
• To demonstrate minimal quantum Cramér-Rao bound for multi-parameter estimation, highlighting NPQC's potential in quantum metrology.
• To enable arbitrary superposition state preparation between two input states using NPQC parameters.

Proposed method

• Design a hardware-efficient NPQC using single-qubit rotations and CPHASE gates, structured so that the QFIM at the reference parameter θ_r is the identity matrix (F(θ_r) = I).
• Leverage the fact that when F(θ) = cI, the standard gradient is equivalent to the quantum natural gradient (QNG), enabling adaptive learning rates without QFIM evaluation.
• Use a first-order Taylor expansion of the state evolution to derive a protocol for estimating absolute parameter values |Δθ| via computational basis measurements.
• Implement the parameter estimation protocol in parallel across all parameters, enabling fast calibration of NISQ devices.
• Construct a parameterized circuit that generates arbitrary superpositions of two input states by solving for the required rotation angles in the NPQC framework.
• Prove that for a general class of PQCs, the NPQC achieves the minimal possible quantum Cramér-Rao bound, indicating optimal estimation precision.

Experimental results

Research questions

• RQ1 Can a parameterized quantum circuit be constructed such that its quantum Fisher information metric is Euclidean at a reference point, enabling natural gradient optimization without QFIM computation?
• RQ2 Can the parameters of such a circuit be estimated efficiently via sampling on NISQ hardware?
• RQ3 Does the NPQC achieve the minimal quantum Cramér-Rao bound in multi-parameter estimation tasks?
• RQ4 Can the NPQC be used to generate arbitrary superpositions of two quantum states?
• RQ5 How does the NPQC’s performance in variational quantum algorithm training compare to standard PQCs, especially in terms of initial convergence speed?

Key findings

• The NPQC achieves a Euclidean quantum geometry at the reference parameter, with F(θ_r) = I, ensuring the standard gradient is equivalent to the quantum natural gradient (QNG).
• Initial training steps in variational quantum algorithms using NPQC are substantially faster because the gradient is equivalent to QNG without requiring QFIM computation.
• The first training step scales favorably with increasing qubit number, indicating potential for scalability.
• The NPQC enables parallel estimation of all parameter magnitudes |Δθ| via computational basis sampling, with accuracy improving for small parameter deviations.
• The NPQC achieves the minimal possible quantum Cramér-Rao bound for a general class of PQCs, indicating optimal precision in multi-parameter estimation.
• The NPQC can generate arbitrary superpositions of two input states by solving for the required rotation parameters, enabling flexible state preparation.