Related papers: The Structure of Sum-Over-Paths, its Consequences,…
The "Sum-Over-Paths" formalism is a way to symbolically manipulate linear maps that describe quantum systems, and is a tool that is used in formal verification of such systems. We give here a new set of rewrite rules for the formalism, and…
The "Sum-Over-Paths" formalism is a way to symbolically manipulate linear maps that describe quantum systems, and is a tool that is used in formal verification of such systems. We give here a new set of rewrite rules for the formalism, and…
Path sums are a convenient symbolic formalism for quantum operations with applications to the simulation, optimization, and verification of quantum protocols. Unlike quantum circuits, path sums are not limited to unitary operations, but can…
We reformulate quantum computation in terms of Lagrangian (sum-over-path) formalism, in contrast to the widely used Hamiltonian (unitary gate) formulation. We exemplify this formalism with some widely-studied models, including the standard…
By the Gottesman-Knill Theorem, the outcome probabilities of Clifford circuits can be computed efficiently. We present an alternative proof of this result for quopit Clifford circuits (i.e., Clifford circuits on collections of $p$-level…
The ZH-calculus is a complete graphical calculus for linear maps between qubits that admits a straightforward encoding of hypergraph states and circuits arising from the Toffoli+Hadamard gate set. In this paper, we establish a…
The notion of overlap algebra introduced by G. Sambin provides a constructive version of complete Boolean algebra. Here we first show some properties concerning overlap algebras: we prove that the notion of overlap morphism corresponds…
Graph transformation theory relies upon the composition of rules to express the effects of sequences of rules. In practice, graphs are often subject to constraints, ruling out many candidates for composed rules. Focusing on the case of…
In this paper we start from a basic notion of process, which we structure into two groupoids, one orthogonal and one symplectic. By introducing additional structure, we convert these groupoids into orthogonal and symplectic Clifford…
Vilmart recently gave a complete equational theory for the balanced sum-over-paths over Toffoli-Hadamard circuits, and by extension Clifford+Rz(2pi/2^k) circuits. Their theory is based on the phase-free ZH-calculus which crucially omits the…
Strongly simulating a quantum circuit, that is, computing an output amplitude, amounts to summing the circuit's Feynman paths, a weighted count over assignments to the Boolean ``path'' variables. The circuit's gates induce correlations…
Motivated by the problem of deformation quantization we introduce and study directed graph complexes with oriented loops and wheels. We develop some technique for computing cohomology of such graph complexes and apply it to several concrete…
Most classical and post-quantum cryptographic assumptions, including integer factorization, discrete logarithms, and Learning with Errors (LWE), rely on algebraic structures such as rings or vector spaces. While mathematically powerful,…
In quantum field theory the path integral is usually formulated in the wave picture, i.e., as a sum over field evolutions. This path integral is difficult to define rigorously because of analytic problems whose resolution may ultimately…
The path decomposition expansion is a path integral technique for decomposing sums over paths in configuration space into sums over paths in different spatial regions. It leads to a decomposition of the configuration space propagator across…
Equivalence checking of quantum circuits is a central verification task in quantum computing, ensuring the correctness of circuit optimizations, hardware mappings, and compilation pipelines. Among the primary symbolic methods for this…
Process calculi and graph transformation systems provide models of reactive systems with labelled transition semantics. While the semantics for process calculi is compositional, this is not the case for graph transformation systems, in…
We formalize the correspondence between quantum states and quantum operations isometrically, and harness its consequences. This correspondence was already implicit in the various proofs of the operator sum representation of Completely…
We use the worldline path-integral approach to the Bern-Kosower formalism for developing a new algorithm for calculation of the sum of all diagrams with one spinor loop and fixed numbers of external and internal photons. The method is based…
The form factor of a quantum graph is a function measuring correlations within the spectrum of the graph. It can be expressed as a double sum over the periodic orbits on the graph. We propose a scheme which allows one to evaluate the…