Circuit-to-Hamiltonian from tensor networks and fault tolerance

TL;DR

This work introduces a clock-free circuit-to-Hamiltonian mapping by encoding quantum circuits into injective tensor networks (PEPS) and associating a local parent Hamiltonian whose unique ground state represents a noisy version of the computation. It develops a fault-tolerant framework that relates low-energy or combinatorial states to adversarially faulted executions of the circuit, yielding both combinatorial and energy-density soundness results and a spectral-gap analysis. These structural insights enable a new proof of the QMA-completeness of the log-local Hamiltonian problem and show that contracting injective tensor networks to additive error is BQP-hard, illuminating potential routes toward a polylogarithmic quantum PCP. The paper also analyzes the computational complexity of injective TNs, discusses fault-tolerance implications, and highlights open questions about robust polylogarithmic PCP-type statements and depth-efficient quantum verification.

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.

Figures (13)

• Figure 1: The circuit $W$ consisting of a collection of gates (black boxes). This layout suffices to implement an arbitrary quantum circuit. However, our construction applies to general circuit layouts.
• Figure 2: Representation of the state $\ket{\Phi^{(\ell)}_{p,q}}$. Qubits 1 and 3, as well as 2 and 4, are in the Bell state $\ket{\Phi_I}$, which is indicated by the blue wavy lines. The unitary $U^{(\ell)}_{p,q}$ is applied to qubits 3 and 4 (black box).
• Figure 3: The circuit $W$ (\ref{['fig:brickCirc']}) converted into a tensor network. We introduce a Bell pair for every position in the circuit (black dots connected by a wavy line) and apply the unitary operation corresponding to the location in the circuit (cf. \ref{['fig:state']}). We then apply projectors on pairs of qubits (gray boxes).
• Figure 4: An injective PEPS encoding noisy quantum computation shown with $n=6$ qubits (black dots), of which $a=3$ are ancillas, and $D=3$ layers of two-qubit gates in the brickwork architecture. The computation goes from left to right, with qubits on column 1 being the input. Gates are encoded in 4-qubit Choi states (see \ref{['fig:state']}) placed on columns (2,3), (4,5), and so on. Applying the invertible map $Q$ (gray box) as defined in \ref{['eqn:perturbed_projectors']} generates a noisy computation on the last column (indexed 7). The qubit pairs where $Q$ is applied are called shifted EPR locations. We refer to the last column of qubits in the PEPS as the output column. Noisy computation: After $Q$ is applied, the output column can be interpreted as a noisy computation where for each layer of the circuit, the present noise pattern is specified by the EPR states at the shifted EPR locations (the word 'shifted' is to avoid confusion with the original locations of the Choi state encodings: the shifted EPR locations are the same as shifting the original Choi state's locations one step to the left). Due to this correspondence, we refer to the first two columns (indexed 1,2) as the first layer, the next two columns (indexed 3,4) as the second layer, and so on. Parent Hamiltonian: A propagation term (dashed green) acts on $8$ qubits, while an initialization term (dashed yellow) acts on the first $2$ qubits and only on each ancilla row (indexed 1,2,3).
• Figure 5: Propagation term $h_j^{(D)} = \Lambda_{AB} (\mathrm{I} - \ket{\Phi_I} \bra{\Phi_I}_{BC}) \Lambda_{AB}$ corresponding to a single-qubit identity gate in the last layer. If a global state $\ket{\psi}$ has low energy with restpect to $h_j^{(D)}$, then \ref{['lem:last-col']} asserts that $V^\dagger \ket{\psi}$ is locally close to $\ket{\phi_0}$ (\ref{['eq:phi0']}) on qubits $A,B$.

Theorems & Definitions (63)

• Claim 2.1
• proof
• Definition 2.2: Parent Hamiltonian
• Claim 2.3
• proof
• Definition 3.1
• Lemma 3.2: Weak $\mathop{\mathrm{\mathsf{QMA}}}\nolimits$ amplification kitaev2002classical
• Lemma 3.3: Strong $\mathop{\mathrm{\mathsf{QMA}}}\nolimits$ amplification marriott2005quantum
• Definition 3.4: $k$-Local Hamiltonian problem
• Theorem 3.5: Kitaev kitaev2002classical