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The Quantum States of a Graph

Mohd Arif Raza, Adel N. Alahmadi, Widyan Basaffar, David G. Glynn, Manish K. Gupta, James W. P. Hirschfeld, Abdul Nadim Khan, Hatoon Shoaib, Patrick Solé

Published May 16, 2023• 1 citation
Publisher

Abstract

Quantum codes are crucial building blocks of quantum computers. With a self-dual quantum code is attached, canonically, a unique stabilised quantum state. Improving on a previous publication, we show how to determine the coefficients on the basis of kets in these states. Two important ingredients of the proof are algebraic graph theory and quadratic forms. The Arf invariant, in particular, plays a significant role.

DOI
10.3390/math11102310

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Quantum codes are crucial building blocks of quantum computers. With a self-dual quantum code is attached, canonically, a unique stabilised quantum state. Improving on a previous publication, we show how to determine the coefficients on the basis of kets in these states. Two important ingredients of the proof are algebraic graph theory and quadratic forms. The Arf invariant, in particular, plays a significant role.
We study inner products and partial amplitudes of graph states--a commonly employed class of quantum states, which are specified by graphs. We find that the magnitudes of these quantities are simply related to the rank of the adjacency matrix of the graph over F_2 while the phase is determined by the Arf invariant of its quadratic refinement. These facts motivate a nonlocal tensor factorization of the Hilbert space, with respect to which all graph states are products of Bell pairs with unentangled ancillae. These results may illuminate the quantum advantage in the framework of Measurement-Based Quantum Computation and suggest that graph states can be usefully visualized in the language of algebraic topology. In addition, we develop a specialized technique for computing expectation values of qubit-wise permutations in graph states, which is useful for calculating multi-invariants.
Symmetry is at the heart of coding theory. Codes with symmetry, especially cyclic codes, play an essential role in both theory and practical applications of classical error-correcting codes. Here we examine symmetry properties for codeword stabilized (CWS) quantum codes, which is the most general framework for constructing quantum error-correcting codes known to date. A CWS code Q can be represented by a self-dual additive code S and a classical code C, i.,e., Q=(S,C), however this representation is in general not unique. We show that for any CWS code Q with certain permutation symmetry, one can always find a self-dual additive code S with the same permutation symmetry as Q such that Q=(S,C). As many good CWS codes have been found by starting from a chosen S, this ensures that when trying to find CWS codes with certain permutation symmetry, the choice of S with the same symmetry will suffice. A key step for this result is a new canonical representation for CWS codes, which is given in terms of a unique decomposition as union stabilizer codes. For CWS codes, so far mainly the standard form (G,C) has been considered, where G is a graph state. We analyze the symmetry of the corresponding graph of G, which in general cannot possess the same permutation symmetry as Q. We show that it is indeed the case for the toric code on a square lattice with translational symmetry, even if its encoding graph can be chosen to be translational invariant.
A short introduction to quantum error correction is given, and it is shown that zero-dimensional quantum codes can be represented as self-dual additive codes over GF(4) and also as graphs. We show that graphs representing several such codes with high minimum distance can be described as nested regular graphs having minimum regular vertex degree and containing long cycles. Two graphs correspond to equivalent quantum codes if they are related by a sequence of local complementations. We use this operation to generate orbits of graphs, and thus classify all inequivalent self-dual additive codes over GF(4) of length up to 12, where previously only all codes of length up to 9 were known. We show that these codes can be interpreted as quadratic Boolean functions, and we define non-quadratic quantum codes, corresponding to Boolean functions of higher degree. We look at various cryptographic properties of Boolean functions, in particular the propagation criteria. The new aperiodic propagation criterion (APC) and the APC distance are then defined. We show that the distance of a zero-dimensional quantum code is equal to the APC distance of the corresponding Boolean function. Orbits of Boolean functions with respect to the {I,H,N}^n transform set are generated. We also study the peak-to-average power ratio with respect to the {I,H,N}^n transform set (PAR_IHN), and prove that PAR_IHN of a quadratic Boolean function is related to the size of the maximum independent set over the corresponding orbit of graphs. A construction technique for non-quadratic Boolean functions with low PAR_IHN is proposed. It is finally shown that both PAR_IHN and APC distance can be interpreted as partial entanglement measures.
Abstract Stabiliser states play a central role in the theory of quantum computation. For example, they are used to encode computational basis states in the most common quantum error correction schemes. Arbitrary quantum states admit many stabiliser decompositions: ways of being expressed as a superposition of stabiliser states. Understanding the structure of stabiliser decompositions has significant applications in verifying and simulating near-term quantum computers. We introduce and study the vector space of linear dependencies of n-qubit stabiliser states. These spaces have canonical bases containing vectors whose size grows exponentially in n. We construct elegant bases of linear dependencies of constant size three. Critically, our sparse bases can be computed without first compiling a dictionary of all n-qubit stabiliser states. We utilise them to explicitly compute the stabiliser extent of states of more qubits than is feasible with existing techniques. Finally, we delineate future applications to improving theoretical bounds on the stabiliser rank of magic states.
Improving the simulation of quantum circuits on classical computers is important for understanding quantum advantage and increasing development speed. In this paper, we explore a way to express stabilizer states and further improve the speed of simulating stabilizer circuits with a current existing approach. First, we discover a unique and elegant canonical form for stabilizer states based on graph states to better represent stabilizer states and show how to efficiently simplify stabilizer states to canonical form. Second, we develop an improved algorithm for graph state stabilizer simulation and establish limitations on reducing the quadratic runtime of applying controlled Pauli ZZZ gates. We do so by creating a simpler formula for combining two Pauli-related stabilizer states into one. Third, to better understand the linear dependence of stabilizer states, we characterize all linearly dependent triplets, revealing symmetries in the inner products. Using our controlled Pauli ZZZ algorithm, we improve runtime for inner product computation from O(n3)O({n}^{3})O(n3) to O(nd2)O(n{d}^{2})O(nd2), where ddd is the maximum degree of the graph encountered during the calculation.
Graphs are closely related to quantum error-correcting codes: every stabilizer code is locally equivalent to a graph code and every codeword stabilized code can be described by a graph and a classical code. For the construction of good quantum codes of relatively large block length, concatenated quantum codes and their generalizations play an important role. We develop a systematic method for constructing concatenated quantum codes based on “graph concatenation,” where graphs representing the inner and outer codes are concatenated via a simple graph operation called “generalized local complementation.” Our method applies to both binary and nonbinary concatenated quantum codes as well as their generalizations.