arXiv subjects

Lia Yeh

Publications and source records attributed to Lia Yeh.

At least 19 recordsLinked to original sources

arXiv2026

Algebraic Structure of Quantum Controlled States and Operators

Quantum control is an important logical primitive of quantum computing programs, and an important concept for equational reasoning in quantum graphical calculi. We show that controlled diagrams in the ZXW-calculus admit rich algebraic structure. The perspective of the higher-order map Ctrl recovers the standard notion of quantum controlled gates, while respecting sequential and parallel composition and multiple-control. In this work, we prove that controlled square matrices form a ring and therefore satisfy powerful rewrite rules. We also show that controlled states form a ring isomorphic to multilinear polynomials. Putting these together, we have completeness for polynomials over same-size square matrices. These properties supply new rewrite rules that make factorisation of arbitrary qubit Hamiltonians achievable inside a single graphical calculus.

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arXiv2026

Transversal AND in Quantum Codes

The AND gate is not reversible$\unicode{x2014}$on qubits. However, it is reversible on qutrits, making it a building block for efficient simulation of qubit computation using qutrits. We first observe that there are multiple two-qutrit Clifford+T unitaries that realize the AND gate with T-count 3, and its generalizations to $n$ qubits with T-count $3n-3$. Our main result is the construction of a novel qutrit $\mathopen{[\![} 6,2,2 \mathclose{]\!]}$ quantum error-correcting code with a transversal implementation of the AND gate. The key insight in our approach is that a symmetric T-depth one circuit decomposition$\unicode{x2014}$composed of a CX circuit, T and T dagger gates, followed by the CX circuit in reverse$\unicode{x2014}$of a given unitary can be interpreted as a CSS code. We can increase the code distance by augmenting the code circuit with additional stabilizers while preserving the logical gate. This results in a code with a "built-in" transversal implementation of the original unitary, which can be further concatenated to attain a $\mathopen{[\![} 48,2,4 \mathclose{]\!]}$ code with the same transversal logical gate. Furthermore, we present several protocols for mixed qubit-qutrit codes which we call Qubit Subspace Codes, and for magic state distillation and injection.

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arXiv2026

A dataflow programming framework for linear optical distributed quantum computing

Photonic systems offer a promising platform for interconnecting quantum processors and enabling scalable, networked architectures. Designing and verifying such architectures requires a unified formalism that integrates linear algebraic reasoning with probabilistic and control-flow structures. In this work, we introduce a graphical framework for distributed quantum computing that brings together linear optics, the ZX-calculus, and dataflow programming. Our language supports the formal analysis and optimization of distributed protocols involving both qubits and photonic modes, with explicit interfaces for classical control and feedforward, all expressed within a synchronous dataflow model with discrete-time dynamics. Within this setting, we classify entangling photonic fusion measurements, show how their induced Pauli errors can be corrected via a novel flow structure for fusion networks, and establish correctness proofs for new repeat-until-success protocols enabling arbitrary fusions. Layer by layer, we construct qubit architectures incorporating practical optical components such as beam splitters, switches, and photon sources, with graphical proofs that they are deterministic and support universal quantum computation. Together, these results establish a foundation for verifiable compilation and automated optimization in networked quantum computing.

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arXiv2025

High schoolers excel at Oxford quantum course using pictorial mathematics

We are at the dawn of the second quantum revolution, where our ability to create and control individual quantum systems is poised to drive transformative advancements in basic science, computation, and everyday life. However, quantum theory has long been conceived as notoriously hard to learn, creating a significant barrier to workforce development, informed decision-making by stakeholders and policymakers, and broader public understanding. This paper is concerned with Quantum Picturalism, a novel visual mathematical language for quantum physics. Originally developed over two decades ago to explore the foundational structure of quantum theory, this rigorous diagrammatic framework has since been adopted in both academia and industry as a powerful tool for quantum computing research and software development. Here, we demonstrate its potential as a transformative educational methodology. We report the findings from a pilot study involving 54 UK high school students, randomly selected from a pool of 734 volunteers across the UK. Despite the absence of advanced mathematical prerequisites, these students demonstrated a strong conceptual grasp of key quantum principles and operations. On an assessment comprising university graduate-level exam questions, participants achieved an 82% pass rate, with 48% obtaining a distinction-level grade. These results pave the way for making quantum more inclusive, lowering traditional cognitive and demographic barriers to quantum learning. This approach has the potential to broaden participation in the field and provide a promising new entry point for stakeholders, future experts, and the general public.

physics.ed-ph↗

arXiv2025

Beyond Penrose tensor diagrams with the ZX calculus: Applications to quantum computing, quantum machine learning, condensed matter physics, and quantum gravity

We introduce the Spin-ZX calculus as an elevation of Penrose's diagrams and associated binor calculus to the level of a formal diagrammatic language. The power of doing so is illustrated by the variety of scientific areas we apply it to: permutational quantum computing, quantum machine learning, condensed matter physics, and quantum gravity. Respectively, we analyse permutational computing transition amplitudes, evaluate barren plateaus for SU(2) symmetric ans\"atze, study properties of AKLT states, and derive the minimum quantised volume in loop quantum gravity. Our starting point is the mixed-dimensional ZX calculus, a purely diagrammatic language that has been proven to be complete for finite-dimensional Hilbert spaces. That is, any equation that can be derived in the Hilbert space formalism, can also be derived in the mixed-dimensional ZX calculus. We embed the Spin-ZX calculus inside the mixed-dimensional ZX calculus, rendering it a quantum information flavoured diagrammatic language for the quantum theory of angular momentum, i.e. SU(2) representation theory. We diagrammatically derive the fundamental spin coupling objects - such as Clebsch-Gordan coefficients, symmetrising mappings between qubits and spin spaces, and spin Hamiltonians - under this embedding. Our results establish the Spin-ZX calculus as a powerful tool for representing and computing with SU(2) systems graphically, offering new insights into foundational relationships and paving the way for new diagrammatic algorithms for theoretical physics.

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arXiv2025

From Fermions to Qubits: A ZX-Calculus Perspective

Mapping fermionic systems to qubits on a quantum computer is often the first step for algorithms in quantum chemistry and condensed matter physics. However, it is difficult to reconcile the many different approaches that have been proposed, such as those based on binary matrices, ternary trees, and stabilizer codes. This challenge is further exacerbated by the many ways to describe them -- transformation of Majorana operators, action on Fock states, encoder circuits, and stabilizers of local encodings -- making it challenging to know when the mappings are equivalent. In this work, we present a graphical framework for fermion-to-qubit mappings that streamlines and unifies various representations through the ZX-calculus. To start, we present the correspondence between linear encodings of the Fock basis and phase-free ZX-diagrams. The commutation rules of scalable ZX-calculus allows us to convert the fermionic operators to Pauli operators under any linear encoding. Next, we give a translation from ternary tree mappings to scalable ZX-diagrams, which not only directly represents the encoder map as a CNOT circuit, but also retains the same structure as the tree. Consequently, we graphically prove that ternary tree transformations are equivalent to linear encodings, a recent result by Chiew et al. The scalable ZX representation moreover enables us to construct an algorithm to directly compute the binary matrix for any ternary tree mapping. Lastly, we present the graphical representation of local fermion-to-qubit encodings. Its encoder ZX-diagram has the same connectivity as the interaction graph of the fermionic Hamiltonian and also allows us to easily identify stabilizers of the encoding.

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arXiv2025

Making the quantum world accessible to young learners through Quantum Picturalism: An experimental study

The educational value of a fully diagrammatic approach in a scientific field has never been explored. We present Quantum Picturalism (QPic), an entirely diagrammatic formalism for all of qubit quantum mechanics. This framework is particularly advantageous for young learners as a novel way to teach key concepts such as entanglement, measurement, and mixed-state quantum mechanics in a math-intensive subject. This eliminates traditional obstacles without compromising mathematical correctness - removing the need for matrices, vectors, tensors, complex numbers, and trigonometry as prerequisites to learning. Its significance lies in that a field as complex as Quantum Information Science and Technology (QIST), for which educational opportunities are typically exclusive to the university level and higher, can be introduced at high school level. In this study, we tested this hypothesis, examining whether QPic reduces cognitive load by lowering complex mathematical barriers while enhancing mental computation and conceptual understanding. The data was collected from an experiment conducted in 2023, whereby 54 high school students (aged 16-18) underwent 16 hours of training spread over eight weeks. The post-assessments illustrated promising outcomes in all three specific areas of focus: (1) whether QPic can alleviate technical barriers in learning QIST, (2) ensures that the content and teaching method are age appropriate, (3) increases confidence and motivation in science and STEM fields. There was a notable success rate in terms of teaching outcomes, with 82% of participants successfully passing an end-of-training exam and 48% achieving a distinction, indicating a high level of performance. The unique testing and training regime effectively reduced the technical barriers typically associated with traditional approaches, as hypothesized.

physics.ed-ph↗

arXiv2024

Fusion and flow: formal protocols to reliably build photonic graph states

Photonics offers a promising platform for implementations of measurement-based quantum computing. Recently proposed fusion-based architectures aim to achieve universality and fault-tolerance. In these approaches, computation is carried out by performing fusion and single-qubit measurements on a resource graph state. The verification of these architectures requires linear algebraic, probabilistic, and control flow structures to be combined in a unified formal language. This paper develops a framework for photonic quantum computing by bringing together linear optics, ZX calculus, and dataflow programming. We characterize fusion measurements that induce Pauli errors and show that they are correctable using a novel flow structure for fusion networks. We prove the correctness of new repeat-until-success protocols for the realization of arbitrary fusions and provide a graph-theoretic proof of universality for linear optics with entangled photon sources. The proposed framework paves the way for the development of compilation algorithms for photonic quantum computing.

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arXiv2024

Exact Synthesis of Multiqutrit Clifford-Cyclotomic Circuits

It is known that the matrices that can be exactly represented by a multiqubit circuit over the Toffoli+Hadamard, Clifford+$T$, or, more generally, Clifford-cyclotomic gate set are precisely the unitary matrices with entries in the ring $\mathbb{Z}[1/2,ζ_k]$, where $k$ is a positive integer that depends on the gate set and $ζ_k$ is a primitive $2^k$-th root of unity. In the present paper, we establish an analogous correspondence for qutrits. We define the multiqutrit Clifford-cyclotomic gate set of degree $3^k$ by extending the classical qutrit gates $X$, $CX$, and $CCX$ with the Hadamard gate $H$ and the $T_k$ gate $T_k=\mathrm{diag}(1,ω_k, ω_k^2)$, where $ω_k$ is a primitive $3^k$-th root of unity. This gate set is equivalent to the qutrit Toffoli+Hadamard gate set when $k=1$, and to the qutrit Clifford+$T_k$ gate set when $k>1$. We then prove that a $3^n\times 3^n$ unitary matrix $U$ can be represented by an $n$-qutrit circuit over the Clifford-cyclotomic gate set of degree $3^k$ if and only if the entries of $U$ lie in the ring $\mathbb{Z}[1/3,ω_k]$.

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arXiv2024

The Focked-up ZX Calculus: Picturing Continuous-Variable Quantum Computation

While the ZX and ZW calculi have been effective as graphical reasoning tools for finite-dimensional quantum computation, the possibilities for continuous-variable quantum computation (CVQC) in infinite-dimensional Hilbert space are only beginning to be explored. In this work, we formulate a graphical language for CVQC. Each diagram is an undirected graph made of two types of spiders: the Z spider from the ZX calculus defined on the reals, and the newly introduced Fock spider defined on the natural numbers. The Z and X spiders represent functions in position and momentum space respectively, while the Fock spider represents functions in the discrete Fock basis. In addition to the Fourier transform between Z and X, and the Hermite transform between Z and Fock, we present exciting new graphical rules capturing heftier CVQC interactions. We ensure this calculus is complete for all of Gaussian CVQC interpreted in infinite-dimensional Hilbert space, by translating the completeness in affine Lagrangian relations by Booth, Carette, and Comfort. Applying our calculus for quantum error correction, we derive graphical representations of the Gottesman-Kitaev-Preskill (GKP) code encoder, syndrome measurement, and magic state distillation of Hadamard eigenstates. Finally, we elucidate Gaussian boson sampling by providing a fully graphical proof that its circuit samples submatrix hafnians.

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arXiv2023

Quantum Picturalism: Learning Quantum Theory in High School

Quantum theory is often regarded as challenging to learn and teach, with advanced mathematical prerequisites ranging from complex numbers and probability theory to matrix multiplication, vector space algebra and symbolic manipulation within the Hilbert space formalism. It is traditionally considered an advanced undergraduate or graduate-level subject. In this work, we challenge the conventional view by proposing "Quantum Picturalism" as a new approach to teaching the fundamental concepts of quantum theory and computation. We establish the foundations and methodology for an ongoing educational experiment to investigate the question "From what age can students learn quantum theory if taught using a diagrammatic approach?". We anticipate that the primary benefit of leveraging such a diagrammatic approach, which is conceptually intuitive yet mathematically rigorous, will be eliminating some of the most daunting barriers to teaching and learning this subject while enabling young learners to reason proficiently about high-level problems. We posit that transitioning from symbolic presentations to pictorial ones will increase the appeal of STEM education, attracting more diverse audience.

physics.ed-ph↗

arXiv2023

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.

quant-ph↗

arXiv2023

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.

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arXiv2023

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.

quant-ph↗

arXiv2023

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.

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arXiv2023

Light-Matter Interaction in the ZXW Calculus

In 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.

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arXiv2023

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.

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arXiv2023

Scaling W state circuits in the qudit Clifford hierarchy

We 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 $d^{\text{th}}$ 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 $\{ |0\rangle , |1\rangle \}$ 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 $O(\text{log }N)$ 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\rangle $-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.

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