ZX-calculus publicationshttp://zxcalculus.com/publications.rssAn up to date list of the newest publications related to the ZX-calculusen-USFri, 09 Jun 2023 14:10:48 GMTrfeed v1.0.0https://github.com/svpino/rfeed/blob/master/README.mdQuantum Circuit Optimization of Arithmetic circuits using ZX Calculushttp://arxiv.org/abs/2306.02264Quantum computing is an emerging technology in which quantum mechanical properties are suitably utilized to perform certain compute-intensive operations faster than classical computers. Quantum algorithms are designed as a combination of quantum circuits that each require a large number of quantum gates, which is a challenge considering the limited number of qubit resources available in quantum computing systems. Our work proposes a technique to optimize quantum arithmetic algorithms by reducing the hardware resources and the number of qubits based on ZX calculus. We have utilised ZX calculus rewrite rules for the optimization of fault-tolerant quantum multiplier circuits where we are able to achieve a significant reduction in the number of ancilla bits and T-gates as compared to the originally required numbers to achieve fault-tolerance. Our work is the first step in the series of arithmetic circuit optimization using graphical rewrite tools and it paves the way for advancing the optimization of various complex quantum circuits and establishing the potential for new applications of the same.Aravind Joshi, Akshara Kairali, Renju Raju, Adithya Athreya, Reena Monica P, Sanjay Vishwakarma and Srinjoy GangulySun, 04 Jun 2023 00:00:00 GMThttp://arxiv.org/abs/2306.02264Light-matter interaction in the ZXW calculushttp://arxiv.org/abs/2306.02114In this paper, we develop a graphical calculus to rewrite photonic circuits involving light-matter interactions and non-linear optical effects. We introduce the infinite ZW calculus, a graphical language for linear operators on the bosonic Fock space which captures both linear and non-linear photonic circuits. This calculus is obtained by combining the QPath calculus, a diagrammatic language for linear optics, and the recently developed qudit ZXW calculus, a complete axiomatisation of linear maps between qudits. It comes with a 'lifting' theorem allowing to prove equalities between infinite operators by rewriting in the ZXW calculus. We give a method for representing bosonic and fermionic Hamiltonians in the infinite ZW calculus. This allows us to derive their exponentials by diagrammatic reasoning. Examples include phase shifts and beam splitters, as well as non-linear Kerr media and Jaynes-Cummings light-matter interaction.Giovanni de Felice, Razin A. Shaikh, Boldizsár Poór, Lia Yeh, Quanlong Wang and Bob CoeckeSat, 03 Jun 2023 00:00:00 GMThttp://arxiv.org/abs/2306.02114Speeding up quantum circuits simulation using ZX-Calculushttp://arxiv.org/abs/2305.02669We present a simple and efficient way to reduce the contraction cost of a tensor network to simulate a quantum circuit. We start by interpreting the circuit as a ZX-diagram. We then use simplification and local complementation rules to sparsify it. We find that optimizing graph-like ZX-diagrams improves existing state of the art contraction cost by several order of magnitude. In particular, we demonstrate an average contraction cost 1180 times better for Sycamore circuits of depth 20, and up to 4200 times better at peak performance.Tristan Cam and Simon MartielThu, 04 May 2023 00:00:00 GMThttp://arxiv.org/abs/2305.02669Scaling W state circuits in the qudit Clifford hierarchyhttp://arxiv.org/abs/2304.12504We identify a novel qudit gate which we call the $\sqrt[d]Z$ gate. This is an alternate generalization of the qutrit $T$ gate to any odd prime dimension $d$, in the $d^\textth$ level of the Clifford hierarchy. Using this gate which is efficiently realizable fault-tolerantly should a certain conjecture hold, we deterministically construct in the Clifford+$\sqrt[d]Z$ gate set, $d$-qubit $W$ states in the qudit $\ |0ångle , |1\n̊gle \$ subspace. For qutrits, this gives deterministic and fault-tolerant constructions for the qubit $W$ state of sizes three with $T$ count 3, six, and powers of three. Furthermore, we adapt these constructions to recursively scale the $W$ state size to arbitrary size $N$, in $O(N)$ gate count and $O(\textlog N)$ depth. This is moreover deterministic for any size qubit $W$ state, and for any prime $d$-dimensional qudit $W$ state, size a power of $d$. For these purposes, we devise constructions of the $ |0\rg̊le $-controlled Pauli $X$ gate and the controlled Hadamard gate in any prime qudit dimension. These decompositions, for which exact synthesis is unknown in Clifford+$T$ for $d > 3$, may be of independent interest.Lia YehTue, 25 Apr 2023 00:00:00 GMThttp://arxiv.org/abs/2304.12504The Algebra for Stabilizer Codeshttp://arxiv.org/abs/2304.10584There is a bijection between odd prime dimensional qudit pure stabilizer states modulo invertible scalars and affine Lagrangian subspaces of finite dimensional symplectic $픽_p$-vector spaces. In the language of the stabilizer formalism, full rank stabilizer tableaus are exactly the bases for affine Lagrangian subspaces. This correspondence extends to an isomorphism of props where the composition of stabilizer circuits becomes the relational composition of affine subspaces and the tensor product becomes the direct sum. In this paper, we extend this correspondence between stabilizer circuits and tableaus to the mixed setting; by regarding stabilizer codes as affine coisotropic susbspaces (again only in odd prime qudit dimension/for qubit CSS codes). We show that by splitting the projector for a stabilizer code we recover the error detection protocol and the error correction protocol with affine classical processing power.Cole ComfortThu, 20 Apr 2023 00:00:00 GMThttp://arxiv.org/abs/2304.10584Towards a generic compilation approach for quantum circuits through resynthesishttp://arxiv.org/abs/2304.08814In this paper, we propose a generic quantum circuit resynthesis approach for compilation. We use an intermediate representation consisting of Paulistrings over Z, I and X, I called a ``mixed ZX-phase polynomial``. From this universal representation, we generate a completely new circuit such that all multi-qubit gates (CNOTs) are satisfying a given quantum architecture. Moreover, we attempt to minimize the amount of generated gates. The proposed algorithms generate fewer CNOTs than similar previous methods on different connectivity graphs ranging from 5-20 qubits. In most cases, the CNOT counts are also lower than Qiskit's. For large circuits, containing >= 100 Paulistrings, our proposed algorithms even generate fewer CNOTs than the TKET compiler. Additionally, we give insight into the trade-off between compilation time and final CNOT count.Arianne Meijer - van de GriendTue, 18 Apr 2023 00:00:00 GMThttp://arxiv.org/abs/2304.08814A ZX-Calculus Approach to Concatenated Graph Codeshttp://arxiv.org/abs/2304.08363Quantum Error-Correcting Codes (QECCs) are vital for ensuring the reliability of quantum computing and quantum communication systems. Among QECCs, stabilizer codes, particularly graph codes, have attracted considerable attention due to their unique properties and potential applications. Concatenated codes, whichcombine multiple layers of quantum codes, offer a powerful technique for achieving high levels of error correction with a relatively low resource overhead. In this paper, we examine the concatenation of graph codes using the powerful and versatile graphical language of ZX-calculus. We establish a correspondence between the encoding map and ZX-diagrams, and provide a simple proof of the equivalence between encoding maps in the Pauli X basis and the graphic operation "generalized local complementation" (GLC) as previously demonstrated in [J. Math. Phys. 52, 022201]. Our analysis reveals that the resulting concatenated code remains a graph code only when the encoding qubits of the same inner code are not directly connected. When they are directly connected, additional Clifford operations are necessary to transform the concatenated code into a graphcode, thus generalizing the results in [J. Math. Phys. 52, 022201]. We further explore concatenated graph codesin different bases, including the examination of holographic codes as concatenated graph codes. Our findings showcase the potential of ZX-calculus in advancing the field of quantum error correction.Zipeng Wu, Song Cheng and Bei ZengMon, 17 Apr 2023 00:00:00 GMThttp://arxiv.org/abs/2304.08363Flow-preserving ZX-calculus rewrite rules for optimisation and obfuscationhttp://arxiv.org/abs/2304.08166In the one-way model of measurement-based quantum computation (MBQC), computation proceeds via measurements on a resource state. So-called flow conditions ensure that the overall computation is deterministic in a suitable sense, with Pauli flow being the most general of these. Computations, represented as measurement patterns, may be rewritten to optimise resource use and for other purposes. Such rewrites need to preserve the existence of flow to ensure the new pattern can still be implemented deterministically. The majority of existing work in this area has focused on rewrites that reduce the number of qubits, yet it can be beneficial to increase the number of qubits for certain kinds of optimisation, as well as for obfuscation. In this work, we introduce several ZX-calculus rewrite rules that increase the number of qubits and preserve the existence of Pauli flow. These rules can be used to transform any measurement pattern into a pattern containing only (general or Pauli) measurements within the XY-plane. We also give the first flow-preserving rewrite rule that allows measurement angles to be changed arbitrarily, and use this to prove that the `neighbour unfusion' rule of Staudacher et al. preserves the existence of Pauli flow. This implies it may be possible to reduce the runtime of their two-qubit-gate optimisation procedure by removing the need to regularly run the costly gflow-finding algorithm.Tommy McElvanney and Miriam BackensMon, 17 Apr 2023 00:00:00 GMThttp://arxiv.org/abs/2304.08166Simple ZX and ZH calculi for arbitrary finite dimensions, via discrete integralshttp://arxiv.org/abs/2304.03310The ZX calculus and the ZH calculus use diagrams to denote and to compute properties of quantum operations, and other multi-linear operators described by tensor networks. These calculi involve 'rewrite rules', which are algebraic manipulations of the tensor networks through transformations of diagrams. The way in which diagrams denote tensor networks is through a semantic map, which assigns a meaning to each diagram in a compositional way. Slightly different semantic maps, which may prove more convenient for one purpose or another (e.g., analysing unitary circuits versus analysing counting complexity), give rise to slightly different rewrite systems. Through a simple application of measure theory on discrete sets, we describe a semantic map for ZX and ZH diagrams for qudits of any dimension D>1, well-suited to represent unitary circuits, and admitting simple rewrite rules. In doing so, we reproduce the 'well-tempered' semantics of [arXiv:2006.02557] for ZX and ZH diagrams in the case D=2. We demonstrate rewrite rules for the 'stabiliser fragment' of the ZX calculus and a 'multicharacter fragment' of the ZH calculus; and demonstrate relationships which would allow the two calculi to be used interoperably as a single 'ZXH calculus'.Niel de Beaudrap and Richard D. P. EastThu, 06 Apr 2023 00:00:00 GMThttp://arxiv.org/abs/2304.03310Picturing counting reductions with the ZH-calculushttp://arxiv.org/abs/2304.02524Counting the solutions to Boolean formulae defines the problem #SAT, which is complete for the complexity class #P. We use the ZH-calculus, a universal and complete graphical language for linear maps which naturally encodes counting problems in terms of diagrams, to give graphical reductions from #SAT to several related counting problems. Some of these graphical reductions, like to #2SAT, are substantially simpler than known reductions via the matrix permanent. Additionally, our approach allows us to consider the case of counting solutions modulo an integer on equal footing. Finally, since the ZH-calculus was originally introduced to reason about quantum computing, we show that the problem of evaluating ZH-diagrams in the fragment corresponding to the Clifford+T gateset, is in $FP^#P$. Our results show that graphical calculi represent an intuitive and useful framework for reasoning about counting problems.Tuomas Laakkonen, Konstantinos Meichanetzidis and John van de WeteringWed, 05 Apr 2023 00:00:00 GMThttp://arxiv.org/abs/2304.02524