Circuit-to-Hamiltonian from tensor networks and fault tolerance

TL;DR

This paper introduces a new way to map quantum circuits to local Hamiltonians using tensor networks, avoiding clock registers, and explores the implications for quantum complexity and fault tolerance.

Contribution

It presents a novel tensor network-based construction for quantum circuit Hamiltonians that is robust to noise and offers new insights into quantum complexity and the quantum PCP conjecture.

Findings

Constructs Hamiltonians from tensor networks without clock registers.

Shows robustness of the ground state encoding under stochastic and adversarial noise.

Establishes BQP-hardness of contracting injective tensor networks to additive error.

Abstract

We define a map from an arbitrary quantum circuit to a local Hamiltonian whose ground state encodes the quantum computation. All previous maps relied on the Feynman-Kitaev construction, which introduces an ancillary `clock register' to track the computational steps. Our construction, on the other hand, relies on injective tensor networks with associated parent Hamiltonians, avoiding the introduction of a clock register. This comes at the cost of the ground state containing only a noisy version of the quantum computation, with independent stochastic noise. We can remedy this - making our construction robust - by using quantum fault tolerance. In addition to the stochastic noise, we show that any state with energy density exponentially small in the circuit depth encodes a noisy version of the quantum computation with adversarial noise. We also show that any `combinatorial state' with energy density polynomially small in depth encodes the quantum computation with adversarial noise. This serves as evidence that any state with energy density polynomially small in depth has a similar property. As applications, we give a new proof of the QMA-completeness of the local Hamiltonian problem (with logarithmic locality) and show that contracting injective tensor networks to additive error is BQP-hard. We also discuss the implication of our construction to the quantum PCP conjecture, combining with an observation that QMA verification can be done in logarithmic depth.