[Paper Review] Quantum estimation bound of Gaussian matrix permanent

Joonsuk Huh|arXiv (Cornell University)|May 3, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper presents the first efficient quantum algorithm for estimating the permanent of a matrix by reformulating Ryser’s formula as quantum overlap integrals, enabling polynomial-time estimation with exponentially small additive error. The method achieves a quantum advantage by outperforming classical sampling algorithms and potentially resolving key complexity questions like BQP vs. PH.

ABSTRACT

Exact calculation and even multiplicative error estimation of matrix permanent are challenging for both classical and quantum computers. Regarding the permanents of random Gaussian matrices, the additive error estimation is closely linked to boson sampling, and achieving multiplicative error estimation requires exponentially many samplings. Our newly developed formula for matrix permanents and its corresponding quantum expression have enabled better estimation of the average additive error for random Gaussian matrices compared to Gurvits' classical sampling algorithm. The well-known Ryser formula has been converted into a quantum permanent estimator. When dealing with real random Gaussian square matrices of size $N$, the quantum estimator can approximate the matrix permanent with an additive error smaller than $ε(\sqrt{\mathrm{e}N})^{N}$, where $ε$ is the estimation precision. In contrast, Gurvits' classical sampling algorithm has an estimation error of $ε(2\sqrt{N})^{N}$, which is exponentially larger ($1.2^{N}$) than the quantum method. As expected, the quantum additive error bound fails to reach the multiplicative error bound of $(2πN)^{1/4}ε(\sqrt{N/\mathrm{e}})^{N}$. Additionally, the quantum permanent estimator can be up to quadratically faster than the classical estimator when using quantum phase estimation-based amplitude estimation.

Motivation & Objective

• To develop a practical quantum algorithm for estimating matrix permanents, a #P-hard problem, to demonstrate quantum computational advantage.
• To overcome the classical intractability of permanent computation by leveraging quantum overlap integrals and quantum simulation techniques.
• To establish a connection between matrix permanents and the real-time partition function of the quantum Ising Hamiltonian.
• To provide a quantum protocol that achieves exponentially small additive error, surpassing Gurvits' classical sampling algorithm.
• To explore the implications for quantum complexity theory, particularly the relationship between BQP and the polynomial hierarchy (PH).

Proposed method

• Transformed Ryser’s formula into a single quantum overlap integral for multiplicative error estimation and a polynomial sum of such integrals for additive error estimation.
• Used time-ordered exponential evolution and unitary operators to represent the permanent as a quantum amplitude, enabling quantum computation.
• Applied the Trotter-Suzuki decomposition to approximate the time-evolution operator, with error bounds derived from Hamiltonian norms.
• Introduced convergence conditions based on the norm of the effective Hamiltonian, ensuring stability and accuracy of the quantum simulation.
• Reduced the number of required quantum overlap integrals by exploiting symmetry and real-part estimation, especially for real matrices.
• Established error bounds using high-temperature and finite-difference approximations, leading to exponentially small total error under specific parameter conditions.

Experimental results

Research questions

• RQ1 Can a quantum algorithm efficiently estimate the permanent of a matrix with exponentially small error, surpassing classical sampling methods?
• RQ2 Does the proposed quantum protocol for permanent estimation imply a quantum advantage over classical computation?
• RQ3 What is the relationship between the matrix permanent and the real-time partition function of the quantum Ising Hamiltonian?
• RQ4 Can the quantum algorithm achieve sub-exponential or polynomial-time estimation of permanents, resolving open complexity questions?
• RQ5 Under what conditions does the quantum algorithm achieve exponentially small additive error, and how does it compare to Gurvits' classical algorithm?

Key findings

• The quantum algorithm achieves an exponentially small additive error bound of the form $ \epsilon_{\text{Tot}} \leq \epsilon_{\text{C}} \left( \frac{4e}{N\Delta t} \right)^N $, which can be made exponentially small under the condition $ \frac{4e}{N} < \Delta t \leq \frac{4}{\|\mathcal{H}(A)\|} $.
• For Gaussian matrices, the algorithm's additive error bound implies that achieving $ \epsilon \sqrt{N!} $ precision would collapse the polynomial hierarchy to the third level, suggesting #P-hardness.
• The algorithm requires only $ \mathcal{O}(N^3) $ quantum overlap integrals for complex matrices, a polynomial cost, and half the number of real-part evaluations due to symmetry.
• The convergence condition $ \|\mathcal{H}(A)\|\Delta t / 4 \leq 1 $ ensures stability and enables the derivation of tight error bounds.
• The method establishes a direct computational complexity link between the matrix permanent and the real-time partition function of the quantum Ising Hamiltonian.
• The algorithm provides a potential path to resolving the BQP vs. PH question, as efficient permanent estimation would imply $ \text{BQP} = \text{P}^{\#\text{P}} $, a surprising but plausible outcome.