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Title: Circuit-to-Hamiltonian from Tensor Networks and Fault Tolerance
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 an application, we 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.  more » « less
Award ID(s):
2013303 2238836
PAR ID:
10525479
Author(s) / Creator(s):
; ;
Publisher / Repository:
ACM
Date Published:
ISBN:
9798400703836
Page Range / eLocation ID:
585 to 595
Subject(s) / Keyword(s):
Circuit-to-Hamiltonian Ground states Tensor networks Quantum fault tolerance Quantum PCP conjecture Quantum complexity theory Quantum information theory
Format(s):
Medium: X
Location:
Vancouver BC Canada
Sponsoring Org:
National Science Foundation
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