Building Qutrit Diagonal Gates from Phase Gadgets

Phase gadgets have proved to be an indispensable tool for reasoning about ZX-diagrams, being used in optimisation and simulation of quantum circuits and the theory of measurement-based quantum computation. In this paper we study phase gadgets for qutrits. We present the flexsymmetric variant of the original qutrit ZX-calculus, which allows for rewriting that is closer in spirit to the original (qubit) ZX-calculus. In this calculus phase gadgets look as you would expect, but there are non-trivial differences in their properties. We devise new qutrit-specific tricks to extend the graphical Fourier theory of qubits, resulting in a translation between the 'additive' phase gadgets and a 'multiplicative' counterpart we dub phase multipliers. This enables us to generalise the qubit notion of multiple-control to qutrits in two ways. The first type is controlling on a single tritstring, while the second type applies the gate a number of times equal to the tritwise multiplication modulo 3 of the control qutrits.We show how both types of control can be implemented for any qutrit Z or X phase gate, ancilla-free, and using only Clifford and phase gates. The first requires a polynomial number of gates and exponentially small phases, while the second requires an exponential number of gates, but constant sized phases. This is interesting, because such a construction is not possible in the qubit setting. As an application of these results we find a construction for emulating arbitrary qubit diagonal unitaries, and specifically find an ancilla-free emulation for the qubit CCZ gate that only requires three single-qutrit non-Clifford gates, provably lower than the four T gates needed for qubits with ancilla.Comment: In Proceedings QPL 2022, arXiv:2311.0837

Constructing all qutrit controlled Clifford+T gates in Clifford+T

For a number of useful quantum circuits, qudit constructions have been found which reduce resource requirements compared to the best known or best possible qubit construction. However, many of the necessary qutrit gates in these constructions have never been explicitly and efficiently constructed in a fault-tolerant manner. We show how to exactly and unitarily construct any qutrit multiple-controlled Clifford+T unitary using just Clifford+T gates and without using ancillae. The T-count to do so is polynomial in the number of controls $k$, scaling as $O(k^{3.585})$. With our results we can construct ancilla-free Clifford+T implementations of multiple-controlled T gates as well as all versions of the qutrit multiple-controlled Toffoli, while the analogous results for qubits are impossible. As an application of our results, we provide a procedure to implement any ternary classical reversible function on $n$ trits as an ancilla-free qutrit unitary using $O(3^n n^{3.585})$ T gates.Comment: 14 page

AUTOMATED, CONTEXT-FREE ASSIGNMENT OF ASYMMETRIC ROTOR MICROWAVE SPECTRA

We present a new algorithm, Robust Automated Assignment of Rigid Rotors (RAARR), for assigning rotational spectra of asymmetric tops. The RAARR algorithm can automatically assign experimental spectra under a broad range of conditions, including spectra comprised of multiple mixture components, in about 100 seconds or less. The RAARR algorithm exploits constraints placed by the conservation of energy to find sets of connected lines in an unassigned spectrum. The highly constrained structure of these sets eliminates all but a handful of plausible assignments for a given set, greatly reducing the number of potential assignments that must be evaluated. We successfully apply our algorithm to automatically assign 15 experimental spectra, including 5 previously unassigned species, without prior estimation of molecular rotational constants. In 9 of the 15 cases, the RAARR algorithm successfully assigns two or more mixture components

PROGRESS MADE TOWARDS CONTEXT-FREE MOLECULAR STRUCTURE DETERMINATION FROM ISOTOPOLOGUE ROTATIONAL SPECTROSCOPY

Recent algorithms\footnote{N.A. Seifert et. al., Journal of Molecular Spectroscopy 312, 13 (2015). }\footnote{L. Yeh, L. Satterthwaite, and D. Patterson, The Journal of Chemical Physics 150, 204122 (2019)} have demonstrated \textit{context-free} assignment of rotational constants---needing no knowledge of the chemical species other than the rotational spectrum. Efforts to date to subsequently determine molecular structure require further information including assignment of singly-substituted isotopologues\footnote{J. Yan and D. Patterson. Submitted to the 75th International Symposium on Molecular Spectroscopy (2020). }\footnote{K. Mayer et. al., Proceedings of the 74th International Symposium on Molecular Spectroscopy (2019).}, mass spectroscopy$^d$, and data mining$^d$\footnote{M. Muckle, A. Mikhonin, D. McDaniel, and/or J. Neill, Proceedings of the 74th International Symposium on Molecular Spectroscopy (2019).}. We investigate two methodologies to resolve sign ambiguities of Kraitchman’s substitution coordinates. The first methodology requires candidate rotational constants of doubly-substituted isotopologues. Given many such candidates, we have worked out how to determine 1) which candidates for singly- and doubly-substituted isotopologues are most probable, and 2) doubly-substituted atoms’ relative position octant. This is realizable given resolution of doubly-substituted species in natural abundance, which is 10-100x order of magnitude above our instrument's present signal to noise.\\ \indent The second methodology requires precision measurement of the electric dipole moments and magnetic g-factors of both the parent and singly-substituted isotopologues. The magnetic g-factor is measured via application of a large magnetic field to our microwave spectrometer, as done by Flygare et. al. (1969). For a 6 carbon molecule, this could be realized given the ability to resolve a $\approx 10^{-5}$ percent difference between magnetic g-factors of the parent and singly-substituted isotopolog species, as well as resolve the electric dipole moment to 4 or 5 significant figures. Improving the capabilities of microwave spectrometers to within these thresholds would therefore enable context-free molecular structure determination

The Qudit ZH-Calculus: Generalised Toffoli+Hadamard and Universality

We introduce the qudit ZH-calculus and show how to generalise all the phase-free qubit rules to qudits. We prove that for prime dimensions d, the phase-free qudit ZH-calculus is universal for matrices over the ring Z[e^2(pi)i/d]. For qubits, there is a strong connection between phase-free ZH-diagrams and Toffoli+Hadamard circuits, a computationally universal fragment of quantum circuits. We generalise this connection to qudits, by finding that the two-qudit |0>-controlled X gate can be used to construct all classical reversible qudit logic circuits in any odd qudit dimension, which for qubits requires the three-qubit Toffoli gate. We prove that our construction is asymptotically optimal up to a logarithmic term. Twenty years after the celebrated result by Shi proving universality of Toffoli+Hadamard for qubits, we prove that circuits of |0>-controlled X and Hadamard gates are approximately universal for qudit quantum computing for any odd prime d, and moreover that phase-free ZH-diagrams correspond precisely to such circuits allowing post-selections.Comment: In Proceedings QPL 2023, arXiv:2308.1548

Composing Conversational Negation

Negation in natural language does not follow Boolean logic and is therefore inherently difficult to model. In particular, it takes into account the broader understanding of what is being negated. In previous work, we proposed a framework for the negation of words that accounts for 'worldly context'. This paper extends that proposal now accounting for the compositional structure inherent in language within the DisCoCirc framework. We compose the negations of single words to capture the negation of sentences. We also describe how to model the negation of words whose meanings evolve in the text.Comment: In Proceedings ACT 2021, arXiv:2211.0110

The Qupit Stabiliser ZX-travaganza: Simplified Axioms, Normal Forms and Graph-Theoretic Simplification

We present a smorgasbord of results on the stabiliser ZX-calculus for odd prime-dimensional qudits (i.e. qupits). We derive a simplified rule set that closely resembles the original rules of qubit ZX-calculus. Using these rules, we demonstrate analogues of the spider-removing local complementation and pivoting rules. This allows for efficient reduction of diagrams to the affine with phases normal form. We also demonstrate a reduction to a unique form, providing an alternative and simpler proof of completeness. Furthermore, we introduce a different reduction to the graph state with local Cliffords normal form, which leads to a novel layered decomposition for qupit Clifford unitaries. Additionally, we propose a new approach to handle scalars formally, closely reflecting their practical usage. Finally, we have implemented many of these findings in DiZX, a new open-source Python library for qudit ZX-diagrammatic reasoning.Comment: 44 pages, lots of figures, accepted to QPL 202

Graphical CSS Code Transformation Using ZX Calculus

In this work, we present a generic approach to transform CSS codes by building upon their equivalence to phase-free ZX diagrams. Using the ZX calculus, we demonstrate diagrammatic transformations between encoding maps associated with different codes. As a motivating example, we give explicit transformations between the Steane code and the quantum Reed-Muller code, since by switching between these two codes, one can obtain a fault-tolerant universal gate set. To this end, we propose a bidirectional rewrite rule to find a (not necessarily transversal) physical implementation for any logical ZX diagram in any CSS code. Then we focus on two code transformation techniques: code morphing, a procedure that transforms a code while retaining its fault-tolerant gates, and gauge fixing, where complimentary codes can be obtained from a common subsystem code (e.g., the Steane and the quantum Reed-Muller codes from the [[15,1,3,3]] code). We provide explicit graphical derivations for these techniques and show how ZX and graphical encoder maps relate several equivalent perspectives on these code-transforming operations.Comment: In Proceedings QPL 2023, arXiv:2308.1548

Completeness for arbitrary finite dimensions of ZXW-calculus, a unifying calculus

The ZX-calculus is a universal graphical language for qubit quantum computation, meaning that every linear map between qubits can be expressed in the ZX-calculus. Furthermore, it is a complete graphical rewrite system: any equation involving linear maps that is derivable in the Hilbert space formalism for quantum theory can also be derived in the calculus by rewriting. It has widespread usage within quantum industry and academia for a variety of tasks such as quantum circuit optimisation, error-correction, and education. The ZW-calculus is an alternative universal graphical language that is also complete for qubit quantum computing. In fact, its completeness was used to prove that the ZX-calculus is universally complete. This calculus has advanced how quantum circuits are compiled into photonic hardware architectures in the industry. Recently, by combining these two calculi, a new calculus has emerged for qubit quantum computation, the ZXW-calculus. Using this calculus, graphical-differentiation, -integration, and -exponentiation were made possible, thus enabling the development of novel techniques in the domains of quantum machine learning and quantum chemistry. Here, we generalise the ZXW-calculus to arbitrary finite dimensions, that is, to qudits. Moreover, we prove that this graphical rewrite system is complete for any finite dimension. This is the first completeness result for any universal graphical language beyond qubits.Comment: 47 pages, lots of figure