[Paper Review] Variational Quantum Algorithms for Trace Distance and Fidelity Estimation

Ranyiliu Chen, Zhixin Song+2 more|arXiv (Cornell University)|Dec 10, 2020

Quantum Computing Algorithms and ArchitectureComputer Science97 references62 citations

TL;DR

This paper proposes Variational Trace Distance Estimation (VTDE) and Variational Fidelity Estimation (VFE), two hybrid quantum-classical algorithms for estimating trace distance and quantum fidelity on near-term quantum devices without prior assumptions on input states. VTDE uses a single ancillary qubit and local measurements to estimate the trace norm via unitary optimization, avoiding barren plateaus; VFE leverages Uhlmann's theorem and state purification to reduce fidelity estimation to a unitary optimization problem, both validated numerically and on IBM superconducting hardware with high accuracy for mixed states.

ABSTRACT

Estimating the difference between quantum data is crucial in quantum computing. However, as typical characterizations of quantum data similarity, the trace distance and quantum fidelity are believed to be exponentially-hard to evaluate in general. In this work, we introduce hybrid quantum-classical algorithms for these two distance measures on near-term quantum devices where no assumption of input state is required. First, we introduce the Variational Trace Distance Estimation (VTDE) algorithm. We in particular provide the technique to extract the desired spectrum information of any Hermitian matrix by local measurement. A novel variational algorithm for trace distance estimation is then derived from this technique, with the assistance of a single ancillary qubit. Notably, VTDE could avoid the barren plateau issue with logarithmic depth circuits due to a local cost function. Second, we introduce the Variational Fidelity Estimation (VFE) algorithm. We combine Uhlmann's theorem and the freedom in purification to translate the estimation task into an optimization problem over a unitary on an ancillary system with fixed purified inputs. We then provide a purification subroutine to complete the translation. Both algorithms are verified by numerical simulations and experimental implementations, exhibiting high accuracy for randomly generated mixed states.

Motivation & Objective

• To develop practical quantum algorithms for estimating trace distance and fidelity between general quantum states on near-term NISQ devices.
• To overcome the exponential classical hardness and quantum complexity of estimating trace distance and fidelity for mixed states.
• To avoid barren plateaus in variational quantum algorithms by using a local cost function in VTDE.
• To enable fidelity estimation for arbitrary mixed states using Uhlmann’s theorem and a purification subroutine.
• To demonstrate high-accuracy estimation via numerical simulations and experimental implementations on IBM superconducting hardware.

Proposed method

• VTDE estimates the trace norm of a Hermitian matrix H by optimizing over unitaries acting on an ancillary qubit, using local measurements to extract spectral information.
• The algorithm uses a cost function based on single-qubit expectation values, ensuring logarithmic-depth circuits and avoiding barren plateaus.
• For VFE, the fidelity is estimated by optimizing over unitaries on an ancillary system, leveraging the freedom in purification and Uhlmann’s theorem.
• A purification subroutine is introduced to prepare purifications of unknown mixed states on NISQ devices, requiring only a few ancillary qubits for low-rank states.
• The algorithms are implemented using parameterized quantum circuits (PQCs) with U3 and CNOT gates, enabling efficient gradient computation via analytical methods.
• Numerical and experimental validation is performed on random mixed states using IBM’s superconducting quantum processor.

Experimental results

Research questions

• RQ1 Can trace distance between two unknown general quantum states be estimated efficiently on near-term quantum devices without prior assumptions on the input states?
• RQ2 How can the trace norm of a Hermitian matrix be estimated using only local measurements and a single ancillary qubit?
• RQ3 Can fidelity estimation for arbitrary mixed states be reduced to a variational optimization problem over unitaries on an ancillary system?
• RQ4 What is the resource cost of state purification for mixed states in the context of variational quantum algorithms?
• RQ5 Does the proposed VTDE algorithm avoid the barren plateau problem in shallow circuits due to its local cost function?

Key findings

• VTDE achieves stable performance across different numbers of positive eigenvalues of ρ − σ, unlike the naive approach which degrades significantly when k differs from the true number of positive eigenvalues.
• The naive VTDE algorithm (nVTDE) requires O(k) optimization rounds and k computational basis states, making it inefficient for large k, especially in the worst case of k = 2^n − 1.
• For pure states, ρ − σ has at most one positive eigenvalue, allowing VTDE to use only a single rank-1 projector, significantly reducing resource cost.
• The VFE algorithm leverages Uhlmann’s theorem and purification freedom to transform fidelity estimation into a unitary optimization problem over an ancillary system.
• Numerical simulations and experimental implementations on IBM’s superconducting device confirm high accuracy for both VTDE and VFE across randomly generated mixed states.
• The purification subroutine requires only a small number of ancillary qubits when the input states are low-rank, making the approach scalable for practical applications.