Normalizer circuits and a Gottesman-Knill theorem for infinite-dimensional systems

Abstract

Normalizer circuits [1,2] are generalized Clifford circuits that act on arbitrary finite-dimensional systems H d1 ⊗ • • •⊗ H dn with a standard basis labeled by the elements of a finite Abelian group G = Z d1 × • • • × Z dn . Normalizer gates implement operations associated with the group G and can be of three types: quantum Fourier transforms, group automorphism gates and quadratic phase gates. In this work, we extend the normalizer formalism [1,2] to infinite dimensions, by allowing normalizer gates to act on systems of the form H ⊗a Z : each factor H Z has a standard basis labeled by integers Z, and a Fourier basis labeled by angles, elements of the circle group T. Normalizer circuits become hybrid quantum circuits acting both on continuous-and discrete-variable systems. We show that infinite-dimensional normalizer circuits can be efficiently simulated classically with a generalized stabilizer formalism for Hilbert spaces associated with groups of the form Z a × T b × Z d1 × • • • × Z dn . We develop new techniques to track stabilizer-groups based on normal forms for group automorphisms and quadratic functions. We use our normal forms to reduce the problem of simulating normalizer circuits to that of finding general solutions of systems of mixed real-integer linear equations [3] and exploit this fact to devise a robust simulation algorithm: the latter remains efficient even in pathological cases where stabilizer groups become infinite, uncountable and noncompact. The techniques developed in this paper might find applications in the study of fault-tolerant quantum computation with superconducting qubits [4,5].

Contextuality as a Resource for Models of Quantum Computation with Qubits

“…We extended earlier results on odd-prime dimensional qudits [4,5] and rebits [6], and thereby completed establishing contextuality as a resource in QCSI in arbitrary prime dimensions. We conjecture that this result generalizes to all composite dimensions [15] (the composite odd case was recently covered after completion of this work [16]) and to algebraic extensions of QCSI models based on normalizer gates [11,[17][18][19][20]. Further, we demonstrated the applicability of our result to a concrete qubit QCSI scheme that does not exhibit state independent contextuality while retaining tomographic completeness.…”

Abelian Hypergroups and Quantum Computation

“…Our approach was inspired by a (fairly unexpected) recently discovered connection by Bermejo-Vega-Lin-Van den Nest [36], between Shor's algorithm and two other foundational results in quantum computation, namely, Gottesman's Pauli stabilizer formalism (PSF) [37], widely used in quantum error correction, and the Gottesman-Knill theorem [37][38][39], which proves the efficient classical simulability of Clifford circuits. In short, the BVLVdN connection states that the quantum algorithms for abelian HSPs belong to a common family of highly structured quantum circuits built of normalizer gates over abelian groups [40][41][42] (quantum Fourier transforms, group automorphism gates, and quadratic phase gates). This fact, combined with the generalized Group Stabilizer Formalism (GSF) for simulating normalizer circuits [40][41][42], was used to prove a sharp no-go theorem for finding new quantum algorithms with the standard abelian group Fourier sampling techniques.…”

“…In short, the BVLVdN connection states that the quantum algorithms for abelian HSPs belong to a common family of highly structured quantum circuits built of normalizer gates over abelian groups [40][41][42] (quantum Fourier transforms, group automorphism gates, and quadratic phase gates). This fact, combined with the generalized Group Stabilizer Formalism (GSF) for simulating normalizer circuits [40][41][42], was used to prove a sharp no-go theorem for finding new quantum algorithms with the standard abelian group Fourier sampling techniques.…”

Encoding a qubit in a molecule

“…As is typical of quantum codes, there is an "easy" subset of all possible operators that aid us in the above tasks in a reasonably fault-tolerant manner. For U 1 -rotors, such normalizer or symplectic operations are generated by certain quadratic functions of the rotors' positions and momenta [129,130].…”