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arXiv:2302.08663 · IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Performance Comparison of the Two Reconstruction Methods for Stabilizer-Based Quantum Secret Sharing

Shogo Chiwaki, Ryutaroh Matsumoto

Published Sep 19, 2023• 1 citation
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Abstract

Stabilizer-based quantum secret sharing has two methods to reconstruct a quantum secret: The erasure correcting procedure and the unitary procedure. It is known that the unitary procedure has a smaller circuit width. On the other hand, it is unknown which method has smaller depth and fewer circuit gates. In this letter, it is shown that the unitary procedure has smaller depth and fewer circuit gates than the erasure correcting procedure which follows a standard framework performing measurements and unitary operators according to the measurements outcomes, when the circuits are designed for quantum secret sharing using the [[5, 1, 3]] binary stabilizer code. The evaluation can be reversed if one discovers a better circuit for the erasure correcting procedure which does not follow the standard framework.

DOI
10.1587/transfun.2023tal0001

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Stabilizer-based quantum secret sharing has two methods to reconstruct a quantum secret: The erasure correcting procedure and the unitary procedure. It is known that the unitary procedure has a smaller circuit width. On the other hand, it is unknown which method has smaller depth and fewer circuit gates. In this letter, it is shown that the unitary procedure has smaller depth and fewer circuit gates than the erasure correcting procedure which follows a standard framework performing measurements and unitary operators according to the measurements outcomes, when the circuits are designed for quantum secret sharing using the [[5, 1, 3]] binary stabilizer code. The evaluation can be reversed if one discovers a better circuit for the erasure correcting procedure which does not follow the standard framework.
Abstract For a quantum secret sharing scheme built from a general quantum stabilizer code, no measurement-free circuit has been known for reconstructing its quantum secrets, except particular classes, such as one proposed by Cleve, Gottesman and Lo. We propose a measurement-free reconstruction circuit of quantum secrets in quantum secret sharing based on stabilizer codes. Our reconstruction circuit has width k+∣J∣k+|J|k+∣J∣ k + | J | and consists of O ( k | J |) one- or two-qudit unitary gates when | J | participants reconstruct k -qudit quantum secrets.
Quantum secret sharing is a cryptographic scheme that enables the secure storage and reconstruction of quantum information. While the theory of secret sharing is mature in its development, relatively few studies have explored the performance of quantum secret sharing on actual devices. In this work, we provide a pedagogical description of encoding and decoding circuits for different secret sharing codes, and we test their performance on IBM’s 127-qubit Brisbane system. We evaluate the quality of the implementation by performing a SWAP test between the decoded state and the ideal one, as well as by estimating how well the code preserves entanglement with a reference system. The results indicate that a ((3,5)) threshold secret sharing scheme and a non-threshold 7-qubit scheme perform similarly based on the SWAP test and entanglement fidelity, with both attaining a roughly 70–75% pass rate on the SWAP test for the reconstructed secret. We also investigate one implementation of a ((2,3)) qutrit threshold scheme and find that it performs the worst of all, which is expected due to the additional number of multi-qubit gate operations needed to encode and decode qutrits. A comparison is also made between schemes using mid-circuit measurement versus delayed-circuit measurement.
Quantum error-correcting codes are used to protect quantum information from decoherence. A raw state is mapped, by an encoding circuit, to a codeword so that the most likely quantum errors from a noisy quantum channel can be removed after a decoding process.A good encoding circuit should have some desired features, such as low depth, few gates, and so on. In this paper, we show how to practically implement an encoding circuit of gate complexity O(n(n - k + c)/ log n) for an [[n, k; c]] quantum stabilizer code with the help of c pairs of maximally-entangled states. For the special case of an [[n, k]] stabilizer code with c = 0, the encoding complexity is O(n(n-k)/ log n), which is previously known to be O(n2/ log n). For c > 0, this suggests that the benefits from shared entanglement come at an additional cost of encoding complexity.Finally we discuss decoding of entanglement-assisted quantum stabilizer codes and extend previously known computational hardness results on decoding quantum stabilizer codes.
The ubiquity of stabilizer circuits in the design and operation of quantum computers makes techniques to verify their correctness essential. The simulation of stabilizer circuits, which aims to replicate their behavior using a classical computer, is known to be efficient and provides a means of testing correctness. However, simulation is limited in its ability to examine the exponentially large space of possible measurement outcomes. We propose a comprehensive set of efficient classical algorithms to fully characterize and exhaustively verify stabilizer circuits with Pauli unitaries conditioned on parities of measurements. We introduce, as a practical characterization, a general form for such circuits and provide an algorithm to find a general form of any stabilizer circuit. We then provide an algorithm for checking the equivalence of stabilizer circuits. When circuits are not equivalent our algorithm suggests modifications for reconciliation. Next, we provide an algorithm that characterizes the logical action of a (physical) stabilizer circuit on an encoded input. All of our algorithms provide relations of measurement outcomes among corresponding circuit representations. Finally, we provide an analytic description of the logical action induced by measuring a stabilizer group, with application in correctness proofs of code-deformation protocols including lattice surgery and code switching.
Qudit codes often inherit the parameters of qubit codes, but not necessarily their correctable erasures. We show that a code encoding one qudit of dimension D into seven such qudits that corrects every erasure Steane's code corrects exists exactly when D is a power of 2; for the qutrit color code, exactly when D is odd. This needs no stabilizer assumption: these codes are, up to local unitaries, stabilizer codes of qubits, respectively prime-power qudits. The two patterns differ in one line, which completes the Fano plane only where 1+1=0. As ideal secret sharing schemes, these codes exist exactly where classical ones do: entanglement does not help, unlike for Euler's 36 officers. The same arithmetic makes the entropy cones of qubit and odd-dimensional stabilizer states incomparable from six parties on. At five, the stabilizer cone is strictly smaller than the quantum linear rank cone, answering an open question.