Low-Overhead Tailoring and Learning of Noise in Graph States

TL;DR

This work establishes a low Clifford hierarchy circuit-based scheme for tailoring and learning noise in (hyper)graph states generated by multiple controlled-Z gates and proposes novel methods for characterizing the noise properties of an entangled state at a lower cost than that of state generation.

Abstract

Graph and hypergraph states are important resource states for realizing universal quantum computation and diverse non-local physical phenomena. However, noise learning in such states is challenging due to their large entanglement and magic. This work establishes a low Clifford hierarchy circuit-based scheme for tailoring and learning noise in (hyper)graph states generated by multiple controlled-Z gates. The key is to convert the noisy input state into a diagonal form and derive the convolution equation of diagonal noise rate, which proves to have a lower resource overhead. Such a diagonal form can be used in real quantum simulation by using the same tailoring technique into the graph state input. We demonstrate that a single-depth Bell measurement is sufficient in our scheme for arbitrary graph states, while noise tailoring can be done via Pauli operations. After that, we suggest the Walsh-Hadamard transform and some approximation method for decoding the noise. We prove that constant sampling and polynomial time complexities are sufficient to guarantee bounded noise estimation error with respect to the $l_2$-norm, which usually requires exponential complexity with existing noise estimation methods. The polynomial complexities are maintained even for the more stringent $l_1$ approximation under the sparse noise assumption. Conclusively, we propose novel methods for characterizing the noise properties of an entangled state at a lower cost than that of state generation. Moreover, compared with state verification methods, we can attain richer information on noise rates and enable noise mitigation, thereby providing guarantees for the preparation of graph states.

Figures (7)

• Figure 1: Schematic diagram of noise tailoring and learning of noisy 3rd-order hypergraph state $\rho$. Two qubit gates of $X^{\mathbf{a}_1\;(\mathbf{a}_2)}_\psi$ can be merged into the feedback section.
• Figure 2: Scaling of the $\max\left\{|c^{(w,s)}_j|\right\}^2$ in Eq. \ref{['eq:method_approx_sol']} as we increase the degree of approximation $(w,s)$.
• Figure 3: Estimation accuracy of $4$-qubit noise learning of third-order complete hypergraph state $\ket{K_4}$. We arbitrarily chose the noisy input state with $p_0\simeq0.819$. For each sampling step, we reconstruct $p'$ via Eq. \ref{['eq:main_exactsolution_fwht']}. The green solid line is the $l_1$-distance between the given step's $p'$ and $p$. The dashed and dash-dotted lines correspond to $16$-components of $p'$.
• Figure 4: The averaged value of $\frac{\|p^{(1)}-p^{(2)}\|_1}{\|\mu^{(1)}-\mu^{(2)}\|_1}$. We randomly chose $\mu^{(1)}_{\mathbf{0}},\mu^{(2)}_{\mathbf{0}}\ge 1-\delta$, and other noise rates are uniformly chosen on the probability simplex (Dirichlet distribution mackay2003). Then we calculate quasi-probabilities $p^{(1)},p^{(2)}$ via $p=\frac{1}{2^n}H\sqrt{H\mu}$. We averaged the rate over $5000$ samples. The x-axis represents the number of qubits $n$.
• Figure 5: Schematic diagrom of $18$-qubit Union Jack state. Each vertex represents $\ket{+}$-state and CCZ-gate acts to each face.

Theorems & Definitions (23)

• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 1: Noise convolution for graph states
• proof : Proof of Thm. \ref{['thm:main_2markovian_eq']}
• Proposition 3
• Proposition 4
• proof
• Proposition 5