Noise Inference by Recycling Test Rounds in Verification Protocols

Abstract

Interactive verification protocols for quantum computations allow to build trust between a client and a service provider, ensuring the former that the instructed computation was carried out faithfully. They come in two variants, one without quantum communication that requires large overhead on the server side to coherently implement quantum-resistant cryptographic primitives, and one with quantum communication but with repetition as the only overhead on the service provider's side. Given the limited number of available qubits on current machines, only quantum communication-based protocols have yielded proof of concepts. In this work, we show that the repetition overhead of protocols with quantum communication can be further mitigated if one examines the task of operating a quantum machine from the service provider's point of view. Indeed, we show that the test rounds data, whose collection is necessary to provide security, can indeed be recycled to perform continuous monitoring of noise model parameters for the service provider. This exemplifies the versatility of these protocols, whose template can serve multiple purposes and increases the interest in considering their early integration into development roadmaps of quantum machines.

Figures (12)

• Figure 1: An example of a single faulty entangling gate in graph state preparation. A graph state $|G\rangle$ is prepared by applying CZ gates to qubits initialized in $|+\rangle$. All CZ gates are assumed ideal except one edge $f=(4, 5)$, whose implementation is modeled as a perfect $\mathrm{CZ}_{(4,5)}$ followed by local depolarizing channels on the qubits it touches.
• Figure 2: Propagation of Pauli errors after the faulty $\mathsf{CZ}_{(4,5)}$ gate. The red boxed Pauli operator indicates the error immediately after the faulty gate, and each panel shows its propagation through the remaining $\mathsf{CZ}$ gates in the circuit. A $\mathsf{X}$ error on qubit $4$ (Top left). A $\mathsf{Z}$ error on qubit $4$ which remains localized on $4$ (Top right). A $\mathsf{Z}$ error on qubit $5$ which remains localized on $5$ (Bottom left). An $\mathsf{X}$ error on qubit $5$, which propagates forward through subsequent $\mathsf{CZ}$ gates, leaving $\mathsf{Z}$ errors on later neighboring qubits $6$ and $7$ while the $\mathsf{X}$ error remains on $5$ (Bottom right).
• Figure 3: Two gate orderings used to isolate depolarizing parameters associated with a central qubit. For an ordered triple $(5,6,7)$ with $(5,6),(6,7)\in E$, the relative order of $\mathsf{CZ}_{(5,6)}$ and $\mathsf{CZ}_{(6,7)}$ is swapped while keeping all other remaining gates fixed. Comparing the corresponding trap statistics yields equations that isolate the depolarizing strengths on the central qubit $6$ for the two incident edges.
• Figure 4: Parallelizing the noise parameter estimation. (Top) Construction of the graph $H$ from the edges of $G$. An ordered triple, e.g. $h$, covers two adjacent edges in $G$. These triples form the vertices $W$ of $H$. There is an edge in $H$ between $h$ and $h'$ as they share vertices in $G$. On the contrary $h$ and $h"$ are not connected by an edge in $H$. As a consequence it is possible to fix the order of the $\mathsf{CZ}$ gates corresponding to the edges covered by $h$ and $h"$ independently, which implies parallelization. (Left) and (Right) show how, in such situation, two orderings are enough to extract $\lambda_{(2,4),4}, \lambda_{(3,4),4}, \lambda_{(5,6),5}, \lambda_{(5,7),5}$.
• Figure 5: Possible construction of the graph $H$ for a 2D cluster-state graph $G$. (Top) The black vertices and edges are that of the 2D cluster-state. Qubits are indexed by their coordinates on the grid. The blue and red sets of qubits are the two triples that need to be considered for constructing the graph $H$ from $G$ at qubit $(3,3)$. Indeed, at qubit $(3,3)$ there are four $\lambda$ parameters to estimate, corresponding to the four edges incident to this qubit. Here, each triple infers two different parameters. This being the case for all qubits, the vertices in $H$ can be partitioned in two sets corresponding to horizontal and vertical triples. ()Bottom) The graph $H$ constructed around the horizontal triple of qubit $(3,3)$ denoted $w_H(3,3)$. The vertex $w_V(3,3)$ corresponds to the vertical triple for qubit $(3,3)$. The other red and blue nodes correspond to the horizontal and vertical triples for the qubits of the graph $G$. The neighborhood of $w_H(3,3)$ is made of all triples that share a qubit with it. This amounts to $w_H(1,3), w_H(2,3), w_H(4,3), w_H(5,3)$ for the horizontal ones and $w_V(2,2), w_V(2,3), w_V(2,4), w_V(3,2), w_V(3,3), w_V(3,4), w_V(4,2), w_V(4,3), w_V(4,4)$. This shows that the degree of $w_H(3,3)$ is 13. By symmetry, with periodic boundary conditions on $G$, all nodes in $H$ have degree 13. This means that the chromatic number is bounded by 14 and that 28 orderings are enough to gather all possible depolarizing parameters for the 2D cluster state. Only the edges incident to $w_H(3,3)$ have been pictured.

Theorems & Definitions (1)

• Theorem 2.1: Security of rVBQC LMKO21verifying