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Directly estimating the fidelity of measurement-based quantum computation

David T. Stephen, Michael Foss-Feig

Abstract

In measurement-based quantum computation (MBQC), quantum circuits are implemented using adaptive measurements on an entangled resource state. In practice, the resource state will always be prepared with some noise, and it is crucial to understand the effect of this noise on the operation of MBQC. Typically, one measures the fidelity of the noisy resource state with the assumption that a high fidelity state means a high fidelity computation. However, the precise relationship between these two fidelities is not known. Here, we derive an expression that equates the average fidelity of the MBQC output state to a certain correlation function evaluated on the noisy resource state. Using this expression, we show that state fidelity provides a tight lower bound on average MBQC fidelity. Conversely, we also find that state fidelity can greatly underestimate average MBQC fidelity, implying that state fidelity is not a good indicator of MBQC performance in general. In response, we formulate an efficient method to directly estimate average MBQC fidelity by measuring the aforementioned correlation function. These results therefore improve our ability to characterize noisy resource states in quantum computers and benchmark MBQC performance.

Directly estimating the fidelity of measurement-based quantum computation

Abstract

In measurement-based quantum computation (MBQC), quantum circuits are implemented using adaptive measurements on an entangled resource state. In practice, the resource state will always be prepared with some noise, and it is crucial to understand the effect of this noise on the operation of MBQC. Typically, one measures the fidelity of the noisy resource state with the assumption that a high fidelity state means a high fidelity computation. However, the precise relationship between these two fidelities is not known. Here, we derive an expression that equates the average fidelity of the MBQC output state to a certain correlation function evaluated on the noisy resource state. Using this expression, we show that state fidelity provides a tight lower bound on average MBQC fidelity. Conversely, we also find that state fidelity can greatly underestimate average MBQC fidelity, implying that state fidelity is not a good indicator of MBQC performance in general. In response, we formulate an efficient method to directly estimate average MBQC fidelity by measuring the aforementioned correlation function. These results therefore improve our ability to characterize noisy resource states in quantum computers and benchmark MBQC performance.
Paper Structure (7 sections, 36 equations, 3 figures)

This paper contains 7 sections, 36 equations, 3 figures.

Table of Contents

  1. Computing $\Omega$ for general resource states
  2. Spectrum of $\Omega$
  3. Largest eigenvalue
  4. Smallest eigenvalue
  5. 1D cluster state example
  6. Recurrence relation for $\Omega$
  7. Spectrum of $\Omega$ for 1D cluster state

Figures (3)

  • Figure 1: Illustration of the measurement-free MBQC operator $\Gamma(\bm{\theta})$ for an arbitrary stabilizer state (here with $|\mathcal{O}| = 2$). The qubits are arranged from top to bottom according to the ordering defined by $\prec$.
  • Figure 2: Examples of $T$-stabilizers (left) and $R$-stabilizers (right) for 1D cluster (top) and 2D cluster state with open boundary conditions raussendorf2003measurement (bottom). Output qubits are highlighted in the last column.
  • Figure 3: Demonstration of the sampling algorithm for a 1D cluster state of length $N=4$. Each fork in the tree corresponds to multiplying by the corresponding $R$-stabilizer or not. Greyed-out paths in the $i$-th step are not allowed because the stabilizer at that point did not act as $X$ on qubit $i$. Sampling starts at the top of the tree and navigates to the bottom, taking each allowed path with equal probability. This shows one branch of the sampling algorithm corresponding to the $T$-stabilizer $XIXZ$; the only other branch in this case contains only the trivial identity stabilizer. Both branches are then weighted by a probability of $1/2^{|\mathcal{M}|} = 1/2$. The vertical bars in each stabilizer represent the division between measured and output qubits.

Theorems & Definitions (3)

  • Remark 1
  • Remark 2
  • Remark 3