[Paper Review] Simulation of low-depth quantum circuits as complex undirected graphical models

Sergio Boixo, Sergei V. Isakov+2 more|arXiv (Cornell University)|Dec 14, 2017

Neural Networks and ApplicationsComputer Science3 references80 citations

TL;DR

The paper maps low-depth universal quantum circuits to complex undirected graphical models and uses variable elimination to compute output probabilities, showing scalability limits and providing numerical benchmarks against prior works.

ABSTRACT

Near term quantum computers with a high quantity (around 50) and quality (around 0.995 fidelity for two-qubit gates) of qubits will approximately sample from certain probability distributions beyond the capabilities of known classical algorithms on state-of-the-art computers, achieving the first milestone of so-called quantum supremacy. This has stimulated recent progress in classical algorithms to simulate quantum circuits. Classical simulations are also necessary to approximate the fidelity of multiqubit quantum computers using cross entropy benchmarking. Here we present numerical results of a classical simulation algorithm to sample universal random circuits, on a single workstation, with more qubits and depth than previously reported. For example, circuits with $5 imes 9$ qubits of depth 37, $7 imes 8$ qubits of depth 27, and $10 imes (κ> 10)$) qubits of depth 19 are all easy to sample. We also show up to what depth the sampling, or estimation of observables, is trivially parallelizable. The algorithm is related to the "Feynmann path" method to simulate quantum circuits. For low-depth circuits, the algorithm scales exponentially in the depth times the smaller lateral dimension, or the treewidth, as explained in Boixo et. al., and therefore confirms the bounds in that paper. In particular, circuits with $7 imes 7$ qubits and depth 40 remain currently out of reach. Follow up work on a supercomputer environment will tighten this bound.

Motivation & Objective

• Motivate near-term quantum supremacy benchmarks using sampling from universal random circuits.
• Develop a classical algorithm that maps quantum circuits to undirected graphical models and enables exact amplitude/probability computation.
• Assess scaling and practical limits of the algorithm on 2D-qubit lattices with CZ and other gates.
• Provide numerical benchmarks comparing runtimes and memory usage to prior state-of-the-art simulations.

Proposed method

• Map a quantum circuit to an undirected graphical model with complex factors.
• Represent gates as factors: diagonal one-qubit gates as one-variable factors; non-diagonal gates as two-variable factors; CZ gates as two-qubit diagonal factors.
• Invoke a Feynman path style formulation where amplitudes are sums over Boolean worldlines encoded as variables b_j^k.
• Use variable elimination (exact inference) to compute amplitudes, with complexity exponential in depth times the smaller lateral dimension (or treewidth).
• Exploit diagonal CZ gates to keep factors localized in the computational basis; implement with TensorFlow (and compare C++ and QuickBB-based orderings).
• Provide numerical experiments on 5x5–10x11 qubit grids with depths up to ~40, reporting time per probability and sampling feasibility.

Experimental results

Research questions

• RQ1 Can low-depth universal random circuits be efficiently sampled using a classical algorithm based on undirected graphical models?
• RQ2 How does the complexity of computing amplitudes/probabilities scale with circuit depth, lateral dimension, and treewidth?
• RQ3 What practical circuit sizes (qubits x depth) become tractable on a single workstation or a cluster for probability estimation and cross-entropy benchmarking?
• RQ4 How do different variable elimination orderings (vertical, treewidth-based) compare in practice for these circuits?
• RQ5 Do the output distributions of such circuits follow Porter-Thomas statistics and what is the entropy of the sampled distributions?

Key findings

• Circuits with 5x9 qubits at depth 40, 7x8 qubits at depth 30, and 10x(κ>10) qubits at depth 19 are easy to sample with about one thousand measurements on a single workstation.
• Cross entropy benchmarking with around one million measurements is feasible on a computer cluster for these circuits.
• The time per output probability grows exponentially with depth and with circuit size, with the exponential cost tied to the treewidth (min(depth×smaller lateral dimension, n)).
• The distribution of 200k sampled probabilities from a 7x8 qubit, depth-30 circuit matches Porter-Thomas exponential statistics, yielding entropy close to theoretical value (log(2^{n})−0.4228).
• Using vertical variable elimination ordering yields efficient, portable TensorFlow-based implementation; treewidth-based ordering can be faster but may require non-standard tooling (e.g., Dask).
• The algorithm can compute amplitudes or probabilities exactly and is amenable to parallelization across machines.