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arXiv:2308.10810 · Advanced Quantum Technologies

Evaluating Pauli Errors on Cluster States by Weighted Distances

Pak Shen Choong, Davide Girolami

Published Apr 18, 2024• 1 citation
arXivHTMLPDF

Abstract

Abstract The problem of evaluating the differences between quantum states before and after being affected by errors encoded in unitary transformations is addressed. Standard distance functions, e.g., the Bures length, are not fully adequate for such a task. Weighted distances are instead the appropriate information measures to quantify the distinguishability of multipartite states. Here, the previously introduced weighted Bures length and the newly defined weighted Hilbert‐Schmidt distance are employed to quantify how much single‐qubit Pauli errors alter cluster states. For both types of weighted distances, it is found that different errors of the same dimension change cluster states in a different way, i.e., their detectability is in general different. Indeed, they transform an ideal cluster state into a state whose weighted distances from the input depends on the specific chosen Pauli rotation, as well as the position of the affected qubit in the graph related to the cluster state. As these features are undetected by using standard distances, the study proves the usefulness of weighted distances to monitor key but elusive properties of many‐body quantum systems.

DOI
10.1002/qute.202300267

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Abstract The problem of evaluating the differences between quantum states before and after being affected by errors encoded in unitary transformations is addressed. Standard distance functions, e.g., the Bures length, are not fully adequate for such a task. Weighted distances are instead the appropriate information measures to quantify the distinguishability of multipartite states. Here, the previously introduced weighted Bures length and the newly defined weighted Hilbert‐Schmidt distance are employed to quantify how much single‐qubit Pauli errors alter cluster states. For both types of weighted distances, it is found that different errors of the same dimension change cluster states in a different way, i.e., their detectability is in general different. Indeed, they transform an ideal cluster state into a state whose weighted distances from the input depends on the specific chosen Pauli rotation, as well as the position of the affected qubit in the graph related to the cluster state. As these features are undetected by using standard distances, the study proves the usefulness of weighted distances to monitor key but elusive properties of many‐body quantum systems.
The quantum state overlap is the textbook measure of the difference between two quantum states. Yet, it is inadequate to compare the complex configurations of many-body systems. The problem is inherited by the widely employed quantum state fidelity and related distances. We introduce the weighted distances, a new class of information-theoretic measures that overcome these limitations. They quantify how hard it is to discriminate between two quantum states of many particles, factoring in the structure of the required measurement apparatus. Therefore, they can be used to evaluate both the theoretical and the experimental performances of complex quantum devices. We also show that the newly defined "weighted Bures length" between the input and output states of a quantum process is a lower bound to the experimental cost of the transformation. The result uncovers an exact quantum limit to our ability to convert physical resources into computational ones.
The unitarity of quantum evolutions implies that an overlap between two initial states does not change in time. This property is commonly believed to explain the apparent lack of state sensitivity in quantum theory, a feature that is prevailing in classical chaotic systems. However, classical state sensitivity is based on a distance between two trajectories in phase space which is a completely different mathematical concept than an overlap between two vectors in Hilbert space. It is possible that state sensitivity in quantum theory can be detected with the help of some special metric. Here we show that the recently introduced Weighted Bures length achieves this task. We numerically investigate a unitary cellular automaton of N interacting qubits and analyze how a single-qubit perturbation affects the evolution of WBL between the unperturbed and perturbed states. We observe a linear growth of WBL if the qubits are arranged into a cyclic graph and an exponential growth if they are arranged into a random bipartite graph.
This paper is devoted to the problem of error detection with quantum codes. We show that it is possible to give a consistent definition of the undetected error event. To prove this, we examine possible problem settings for quantum error detection. Our goal is to derive a functional that describes the probability of undetected error under natural physical assumptions concerning transmission with error detection with quantum codes. We discuss possible transmission protocols with stabilizer and unrestricted quantum codes. The set of results proved in the paper shows that in all the cases considered the average probability of undetected error for a given code is essentially given by one and the same function of its weight enumerators. We examine polynomial invariants of quantum codes and show that coefficients of Rains's (see ibid., vol44, p.1388-94, 1998) "unitary weight enumerators" are known for classical codes under the name of binomial moments of the distance distribution. As in the classical situation, these enumerators provide an alternative expression for the probability of undetected error.
We explore basic necessary protocols to achieve fault tolerance for quantum computation in the cluster state and circuit models. For the cluster state model we simulate the implementation of an arbitrary rotation via only measurement on a decohered cluster state. Fidelity is used to quantify the accuracy of the initial cluster state and a gate fidelity for the arbitrary rotation is determined. In the circuit model we compare the accuracy of two methods that can be used to construct a logical zero state appropriate for the [7, 1, 3] Steane quantum error correction code in a non-equiprobable Pauli operator error environment: a fault tolerant method done by applying error correction on seven qubits all in the state zero, and a non-fault tolerant method done by implementing the encoding gate sequence. We find that the latter construction method, in spite of its lack of "fault tolerance," outputs a seven qubit state with a higher fidelity than the first (fault tolerant) method. However, the fidelity of the single qubit of stored information exhibits almost equivalent values between the two construction methods.
Graph states are multiparticle entangled states that correspond to mathematical graphs, where the vertices of the graph take the role of quantum spin systems and edges represent Ising interactions. They are many-body spin states of distributed quantum systems that play a significant role in quantum error correction, multiparty quantum communication, and quantum computation within the framework of the one-way quantum computer. We characterize and quantify the genuine multiparticle entanglement of such graph states in terms of the Schmidt measure, to which we provide upper and lower bounds in graph theoretical terms. Several examples and classes of graphs will be discussed, where these bounds coincide. These examples include trees, cluster states of different dimensions, graphs that occur in quantum error correction, such as the concatenated [7,1,3]-CSS code, and a graph associated with the quantum Fourier transform in the one-way computer. We also present general transformation rules for graphs when local Pauli measurements are applied, and give criteria for the equivalence of two graphs up to local unitary transformations, employing the stabilizer formalism. For graphs of up to seven vertices we provide complete characterization modulo local unitary transformations and graph isomorphisms.
Abstract Quantum state clustering provides a powerful tool for quantum information processing. This paper focuses on the classification of single-qubit pure and mixed states via clustering algorithms, and discovers that the classical K -means algorithm has two limitations in clustering quantum states: the Euclidean distance cannot accurately reflect the similarity between quantum states, and the centroid update based on the arithmetic mean frequently fails to yield valid quantum states. To address these issues, we propose two distance metrics for quantum states: the dynamic distance and the Bloch distance, and adopt the K -medoids method to update the centroid, ensuring that the cluster center always remains a valid quantum state, thereby designing a quantum-data-adaptive clustering algorithm. Experimental results show that the dynamic distance achieves a clustering accuracy as high as 91.4 % , but at the cost of higher computational complexity. In contrast, the Bloch distance significantly reduces computational complexity while maintaining clustering performance comparable to that of the dynamic distance, achieving a clustering accuracy of 87.4 % . This paper provides a practical method for achieving efficient and reliable clustering of single-qubit quantum states.