Large-scale quantum computation is likely to require massive quantum error correction (QEC). QEC codes and circuits are described via the stabilizer formalism, which represents stabilizer states by keeping track of the operators that preserve them. Such states are obtained by stabilizer circuits (consisting of CNOT, Hadamard and Phase gates) and can be represented compactly on conventional computers using Θ(n2) bits, where n is the number of qubits [18]. As an additional application, the work in [1, 2] suggests the use of superpositions of stabilizer states to represent arbitrary quantum states. To aid in such applications and improve our understanding of stabilizer states, we characterize and count nearest-neighbor stabilizer states, quantify the distribution of angles between pairs of stabilizer states, study succinct stabilizer superpositions and stabilizer bivectors, explore the approximation of non-stabilizer states by single stabilizer states and short linear combinations of stabilizer states, develop an improved inner-product computation for stabilizer states via synthesis of compact canonical stabilizer circuits, propose an orthogonalization procedure for stabilizer states, and evaluate several of these algorithms empirically.
Large-scale quantum computation is likely to require massive quantum error correction (QEC). QEC codes and circuits are described via the stabilizer formalism, which represents stabilizer states by keeping track of the operators that preserve them. Such states are obtained by stabilizer circuits (consisting of CNOT, Hadamard and Phase gates) and can be represented compactly on conventional computers using Θ(n2) bits, where n is the number of qubits [18]. As an additional application, the work in [1, 2] suggests the use of superpositions of stabilizer states to represent arbitrary quantum states. To aid in such applications and improve our understanding of stabilizer states, we characterize and count nearest-neighbor stabilizer states, quantify the distribution of angles between pairs of stabilizer states, study succinct stabilizer superpositions and stabilizer bivectors, explore the approximation of non-stabilizer states by single stabilizer states and short linear combinations of stabilizer states, develop an improved inner-product computation for stabilizer states via synthesis of compact canonical stabilizer circuits, propose an orthogonalization procedure for stabilizer states, and evaluate several of these algorithms empirically.
Large-scale quantum computation is likely to require massive quantum error correction (QEC). QEC codes and circuits are described via the stabilizer formalism, which represents stabilizer states by keeping track of the operators that preserve them. Such states are obtained by {\em stabilizer circuits} (consisting of CNOT, Hadamard and Phase gates) and can be represented compactly on conventional computers using Θ(n2) bits, where n is the number of qubits \cite{Gottes98}. As an additional application, the work in~\cite{Aaron, AaronGottes} suggests the use of superpositions of stabilizer states to represent arbitrary quantum states. To aid in such applications and improve our understanding of stabilizer states, we characterize and count nearest-neighbor stabilizer states, quantify the distribution of angles between pairs of stabilizer states, study succinct stabilizer superpositions and stabilizer bivectors, explore the approximation of non-stabilizer states by single stabilizer states and short linear combinations of stabilizer states, develop an improved inner-product computation for stabilizer states via synthesis of compact canonical stabilizer circuits, propose an orthogonalization procedure for stabilizer states, and evaluate several of these algorithms empirically.
Large-scale quantum computation is likely to require massive quantum error correction (QEC). QEC codes and circuits are described via the stabilizer formalism, which represents stabilizer states by keeping track of the operators that preserve them. Such states are obtained by stabilizer circuits (consisting of CNOT, Hadamard and Phase gates) and can be represented compactly on conventional computers using O(n2) bits, where n is the number of qubits. As an additional application, the work by Aaronson and Gottesman suggests the use of superpositions of stabilizer states to represent arbitrary quantum states. To aid in such applications and improve our understanding of stabilizer states, we characterize and count nearest-neighbor stabilizer states, quantify the distribution of angles between pairs of stabilizer states, study succinct stabilizer superpositions and stabilizer bivectors, explore the approximation of non-stabilizer states by single stabilizer states and short linear combinations of stabilizer states, develop an improved inner-product computation for stabilizer states via synthesis of compact canonical stabilizer circuits, propose an orthogonalization procedure for stabilizer states, and evaluate several of these algorithms empirically.
Large-scale quantum computation is likely to require massive quantum error correction (QEC). QEC codes and circuits are described via the stabilizer formalism, which represents stabilizer states by keeping track of the operators that preserve them. Such states are obtained by stabilizer circuits (consisting of CNOT, Hadamard and Phase only) and can be represented compactly on conventional computers using Omega(n^2) bits, where n is the number of qubits. Although techniques for the efficient simulation of stabilizer circuits have been studied extensively, techniques for efficient manipulation of stabilizer states are not currently available. To this end, we design new algorithms for: (i) obtaining canonical generators for stabilizer states, (ii) obtaining canonical stabilizer circuits, and (iii) computing the inner product between stabilizer states. Our inner-product algorithm takes O(n^3) time in general, but observes quadratic behavior for many practical instances relevant to QECC (e.g., GHZ states). We prove that each n-qubit stabilizer state has exactly 4(2^n - 1) nearest-neighbor stabilizer states, and verify this claim experimentally using our algorithms. We design techniques for representing arbitrary quantum states using stabilizer frames and generalize our algorithms to compute the inner product between two such frames.
Stabilizer states hold a special place in quantum information science due to their connection with quantum error correction and quantum circuit simulation. In the context of classical simulations of many-body physics, they are an example of states that can be both highly entangled and efficiently represented and transformed under Clifford operators. Recently, Clifford operators have been discussed in the context of fermionic quantum computation through their extension, the Majorana Clifford group. Here, we document the phase-sensitive form of the corresponding Majorana stabilizer states, as well as the algorithms for computing their amplitudes, their inner products, and update rules for transforming Majorana stabilizer states under Majorana Clifford gates.
Simulation of quantum information processing remains a major challenge with important applications in quantum computer science and engineering. Generic quantum-circuit simulation appears intractable for conventional computers and may be unnecessary because useful quantum circuits exhibit significant structure that can be exploited during simulation. For example, Gottesman and Knill identified an important subclass, called stabilizer circuits, which can be simulated efficiently using the Heisenberg representation for quantum computers. Stabilizer circuits are exclusively composed of stabilizer gates -- Hadamard, Phase and CNOT -- followed by one-qubit measurements in the computational basis. Such circuits are applied to a computational-basis state and produce so-called stabilizer states. Aaronson and Gottesman generalized stabilizer-circuit simulation to additionally handle a small number of non-stabilizer gates. We design new, more efficient data structures and algorithms for such beyond-stabilizer simulation using superpositions of stabilizer states. One such data structure, a stabilizer frame, offers more compact storage than previous approaches but require additional algorithms to maintain the global phases of each state in the superposition. To explore the advantages and limitations of our technique, we analyze the geometric structure of stabilizer states and their embedding in Hilbert space. Our analysis includes results on the computational geometry of stabilizer states such as efficient algorithms for computing distances, angles and volumes between them. The main advantages of using stabilizer-state superpositions to simulate quantum circuits are: (i) stabilizer subcircuits are simulated with high efficiency, (ii) superpositions can be restructured and compressed on the fly during simulation to reduce resource requirements, and (iii) operations performed on such superpositions lend themselves to distributed or asynchronous processing. Our software implementation, called Quipu, simulates certain quantum arithmetic circuits (e.g., reversible ripple-carry adders) and quantum Fourier transform circuits in polynomial time and space for specific input states. On such instances, known linear-algebraic simulation techniques, such as the (state-of-the-art) BDD-based simulator QuIDDPro, take exponential time. We simulate quantum fault-tolerant circuits using Quipu, and the results indicate that our stabilizer-based technique empirically outperforms QuIDDPro in all cases. While previous structure-aware simulations of quantum circuits were difficult to parallellize, we demonstrate a parallel version of Quipu that achieves a nontrivial speedup.
Quantum computing is an emerging technology that has the potential to achieve exponential speedups over their classical counterparts. To achieve quantum advantage, quantum principles are being applied to fields such as communications, information processing, and artificial intelligence. However, quantum computers face a fundamental issue since quantum bits are extremely noisy and prone to decoherence. Keeping qubits error free is one of the most important steps towards reliable quantum computing. Different stabilizer codes for quantum error correction have been proposed in past decades and several methods have been proposed to import classical error correcting codes to the quantum domain. Design of encoding and decoding circuits for the stabilizer codes have also been proposed. Optimization of these circuits in terms of the number of gates is critical for reliability of these circuits. In this paper, we propose a procedure for optimization of encoder circuits for stabilizer codes. Using the proposed method, we optimize the encoder circuit in terms of the number of 2-qubit gates used. The proposed optimized eight-qubit encoder uses 18 CNOT gates and 4 Hadamard gates, as compared to 14 single qubit gates, 33 2-qubit gates, and 6 CCNOT gates in a prior work. The encoder and decoder circuits are verified using IBM Qiskit. We also present encoder circuits for the Steane code and a 13-qubit code, that are optimized with respect to the number of gates used, leading to a reduction in number of CNOT gates by 1 and 8, respectively.