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Frontiers of Physics

Lecture notes on quantum entanglement: From stabilizer states to stabilizer channels

Amir R. Arab

Published Apr 16, 2024• 3 citations
Publisher

Abstract

We study mathematical, physical and computational aspects of the stabilizer formalism arising in quantum information and quantum computation. The measurement process of Pauli observables with its algorithm is given. It is shown that to detect genuine entanglement we need a full set of stabilizer generators and the stabilizer witness is coarser than the GHZ (Greenberger–Horne–Zeilinger) witness. We discuss stabilizer codes and construct a stabilizer code from a given linear code. We also discuss quantum error correction, error recovery criteria and syndrome extraction. The symplectic structure of the stabilizer formalism is established and it is shown that any stabilizer code is unitarily equivalent to a trivial code. The structure of graph codes as stabilizer codes is identified by obtaining the respective stabilizer generators. The distance of embeddable stabilizer codes in lattices is obtained. We discuss the Knill–Gottesman theorem, tableau representation and frame representation. The runtime of simulating stabilizer gates is obtained by applying stabilizer matrices. Furthermore, an algorithm for updating global phases is given. Resolution of quantum channels into stabilizer channels is shown. We discuss capacity achieving codes to obtain the capacity of the quantum erasure channel. Finally, we discuss the shadow tomography problem and an algorithm for constructing classical shadow is given.

DOI
10.1007/s11467-024-1397-4

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We study mathematical, physical and computational aspects of the stabilizer formalism arising in quantum information and quantum computation. The measurement process of Pauli observables with its algorithm is given. It is shown that to detect genuine entanglement we need a full set of stabilizer generators and the stabilizer witness is coarser than the GHZ (Greenberger–Horne–Zeilinger) witness. We discuss stabilizer codes and construct a stabilizer code from a given linear code. We also discuss quantum error correction, error recovery criteria and syndrome extraction. The symplectic structure of the stabilizer formalism is established and it is shown that any stabilizer code is unitarily equivalent to a trivial code. The structure of graph codes as stabilizer codes is identified by obtaining the respective stabilizer generators. The distance of embeddable stabilizer codes in lattices is obtained. We discuss the Knill–Gottesman theorem, tableau representation and frame representation. The runtime of simulating stabilizer gates is obtained by applying stabilizer matrices. Furthermore, an algorithm for updating global phases is given. Resolution of quantum channels into stabilizer channels is shown. We discuss capacity achieving codes to obtain the capacity of the quantum erasure channel. Finally, we discuss the shadow tomography problem and an algorithm for constructing classical shadow is given.
We investigate how stabilizer theory can be used for constructing sufficient conditions for entanglement. First, we show how entanglement witnesses can be derived for a given state, provided some stabilizing operators of the state are known. These witnesses require only a small effort for an experimental implementation and are robust against noise. Second, we demonstrate that also nonlinear criteria based on uncertainty relations can be derived from stabilizing operators. These criteria can sometimes improve the witnesses by adding nonlinear correction terms. All our criteria detect states close to Greenberger-Horne-Zeilinger states, cluster and graph states. We show that similar ideas can be used to derive entanglement conditions for states which do not fit the stabilizer formalism, such as the three-qubit WWW state. We also discuss connections between the witnesses and some Bell inequalities.
Quantum computers need to be protected by quantum error-correcting codes against decoherence. One of the most interesting and useful classes of quantum codes is the class of quantum stabilizer codes. Entanglement-assisted (EA) quantum codes are a class of stabilizer codes that make use of preshared entanglement between the sender and the receiver. We provide several code constructions for entanglement-assisted quantum codes. The MacWilliams identity for quantum codes leads to linear programming bounds on the minimum distance. We find new constraints on the simplified stabilizer group and the logical group, which help improve the linear programming bounds on entanglement-assisted quantum codes. The results also can be applied to standard stabilizer codes. In the real world, quantum gates are faulty. To implement quantum computation fault-tolerantly, quantum codes with certain properties are needed. We first analyze Knill's postselection scheme in a two-dimensional architecture. The error performance of this scheme is better than other known concatenated codes. Then we propose several methods to protect syndrome extraction against measurement errors.
The quantum hashing bound guarantees that rates up to 1−H(pI,pX,pY,pZ)1-H(p_I, p_X, p_Y, p_Z)1−H(pI​,pX​,pY​,pZ​) are achievable for memoryless Pauli channels, but it is not generally tight. A known way to improve achievable rates for certain asymmetric Pauli channels is to apply a small inner stabilizer code to a few channel uses, decode, and treat the resulting logical noise as an induced Pauli channel; reapplying the hashing argument to this induced channel can beat the baseline hashing bound. We generalize this induced-channel viewpoint to arbitrary stabilizer codes used purely as channel transforms. Given any $ [\![ n, k ]\!] $ stabilizer generator set, we construct a full symplectic tableau, compute the induced joint distribution of logical Pauli errors and syndromes under the physical Pauli channel, and obtain an achievable rate via a hashing bound with decoder side information. We perform a structured search over small transforms and report instances that improve the baseline hashing bound for a family of Pauli channels with skewed and independent errors studied in prior work.
The stabilizer formalism is a central framework in quantum computing and one of the key building blocks for fault-tolerant quantum computers. According to the Gottesman–Knill theorem, we can efficiently simulate any quantum system consisting of stabilizer states, Clifford gates, and Pauli measurements on a classical computer. A key challenge in developing robust quantum technologies is that physical quantum systems are unavoidably affected by noise, which degrades quantum information. Accurate noise modeling is therefore essential. For stabilizer states, the noisy stabilizer formalism has recently been introduced as a framework for efficiently describing the evolution of graph states subject to Pauli-diagonal noise under Clifford operations and Pauli measurements. In this thesis, we explore and extend this formalism in two directions. First, we generalize the method from graph states to arbitrary stabilizer states, thereby extending the Gottesman-Knill theorem to noisy scenarios. Second, we extend the formalism to handle noise models beyond Pauli-diagonal channels. Moreover, we study the applicability of the noisy stabilizer formalism to entanglement purification protocols.
Long distance communication of digital data, whether through a physical medium or a broadcast signal, is often subjected to noise. To deliver data reliably through noisy communication channels, one must use codes that can detect and correct the particular noise of the channel. For transmission of classical data, error correcting schemes can be as simple as the sending of replicates. For quantum data, and in tandem the development of machines that can process quantum data, quantum error correcting codes must be developed. In addition to a larger set of possible errors, quantum error correcting schemes must contend with other peculiarities of quantum mechanics, such as the no-cloning theorem which can prevent the sending of replicate messages. Stabilizer codes are one family of quantum error correcting codes which can protect and correct errors expressed in terms of the Pauli group, exploiting its group structure and utilizing classical codes and the corresponding duals. We develop and examine a family of quantum stabilizer codes which arise from reflexive stabilizers. Moreover, we provide a mapping from our reflexive stabilizer codes to the well-known CSS codes developed by Calderbank, Shor, and Steane. For the case of a 4-state system we show that these codes can obtain the minimal embedding for code which can correct any flip or phase error. We also provide heuristic algorithms for creating reflexive stabilizer codes starting from the noise of a quantum channel. Furthermore, we show that the problem can be posed in terms of finding maximal Cayley subgraphs with restrictions imposed by the set of potential errors.
The extrinsic information transfer (EXIT) and generalized EXIT (GEXIT) area theorems provide fundamental conservation laws that connect the rate of classical codes to the performance of maximum-a-posteriori decoding on binary memoryless channels. In this work, we generalize these area theorems to quantum error correction. We first develop a coset-entropy formalism for nested linear codes, where the message specifies a coset and encoder randomization selects a codeword within that coset. The area under the resulting coset (G)EXIT function equals the normalized dimension of the quotient. For stabilizer codes, the measured syndrome restricts the Pauli error but does not always determine its logical effect, and the remaining uncertainty is an entropy over logical correction classes. We show that this entropy equals a classical coset entropy associated with the stabilizer group and the Pauli operators that commute with it. The resulting quantum GEXIT function has area equal to the number of encoded qubits per channel use for any stabilizer code, and this applies to the depolarizing channel through its joint four-valued Pauli-error distribution. For CSS codes on the quantum erasure channel and under independent bit- and phase-flip noise, the logical correction problem separates into two binary coset-decoding problems. On the quantum erasure channel, the duality between the stabilizer group and the Pauli operators that commute with it gives a folding identity at complementary erasure probabilities. Using this and the area theorem, we derive an upper bound on the rate of code families with vanishing logical error probability that matches the capacity of the quantum erasure channel. For the depolarizing channel, we also derive a rate-dependent converse, using a connection between correction-class entropy and coherent information to establish an inequality involving the complementary channel.