Abstract Pseudo-unitary circuits are recurring in both S-matrix theory and analysis of No-Go theorems. We propose a matrix and diagrammatic representation for the operation that maps S-matrices to T-matrices and, consequently, a unitary group to a pseudo-unitary one. We call this operation ‘partial inversion’ and show its diagrammatic representation in terms of permutations. We find the expressions for the deformed metrics and deformed dot products that preserve physical constraints after partial inversion. Subsequently, we define a special set that allows for the simplification of expressions containing infinities in matrix inversion. Finally, we propose a renormalized-growth algorithm for the T-matrix as a possible application. The outcomes of our study expand the methodological toolbox needed to build a family of pseudo-unitary and inter-pseudo-unitary circuits with full diagrammatic representation in three dimensions, so that they can be used to exploit pseudo-unitary flexibilization of unitary No-Go theorems and renormalized circuits of large scattering lattices.
Abstract Pseudo-unitary circuits are recurring in both S-matrix theory and analysis of No-Go theorems. We propose a matrix and diagrammatic representation for the operation that maps S-matrices to T-matrices and, consequently, a unitary group to a pseudo-unitary one. We call this operation ‘partial inversion’ and show its diagrammatic representation in terms of permutations. We find the expressions for the deformed metrics and deformed dot products that preserve physical constraints after partial inversion. Subsequently, we define a special set that allows for the simplification of expressions containing infinities in matrix inversion. Finally, we propose a renormalized-growth algorithm for the T-matrix as a possible application. The outcomes of our study expand the methodological toolbox needed to build a family of pseudo-unitary and inter-pseudo-unitary circuits with full diagrammatic representation in three dimensions, so that they can be used to exploit pseudo-unitary flexibilization of unitary No-Go theorems and renormalized circuits of large scattering lattices.
We study representation theory of the partially transposed permutation matrix algebra, a matrix representation of the diagrammatic walled Brauer algebra. This algebra plays a prominent role in mixed Schur-Weyl duality that appears in various contexts in quantum information. Our main technical result is an explicit formula for the action of the walled Brauer algebra generators in the Gelfand-Tsetlin basis. It generalizes the well-known Gelfand-Tsetlin basis for the symmetric group (also known as Young's orthogonal form or Young-Yamanouchi basis). We provide two applications of our result to quantum information. First, we show how to simplify semidefinite optimization problems over unitary-equivariant quantum channels by performing a symmetry reduction. Second, we derive an efficient quantum circuit for implementing the optimal port-based quantum teleportation protocol, exponentially improving the known trivial construction. As a consequence, this also exponentially improves the known lower bound for the amount of entanglement needed to implement unitaries non-locally. Both applications require a generalization of quantum Schur transform to tensors of mixed unitary symmetry. We develop an efficient quantum circuit for this mixed quantum Schur transform and provide a matrix product state representation of its basis vectors. For constant local dimension, this yields an efficient classical algorithm for computing any entry of the mixed quantum Schur transform unitary.
Quantum R-matrices, the entangling deformations of non-entangling (classical) permutations, provide a distinguished basis in the space of unitary evolutions and, consequently, a natural choice for a minimal set of basic operations (universal gates) for quantum computation. Yet they play a special role in group theory, integrable systems and modern theory of non-perturbative calculations in quantum field and string theory. Despite recent developments in those fields the idea of topological quantum computing and use of R-matrices, in particular, practically reduce to reinterpretation of standard sets of quantum gates, and subsequently algorithms, in terms of available topological ones. In this paper we summarize a modern view on quantum R-matrix calculus and propose to look at the R-matrices acting in the space of irreducible representations, which are unitary for the real-valued couplings in Chern–Simons theory, as the fundamental set of universal gates for topological quantum computer. Such an approach calls for a more thorough investigation of the relation between topological invariants of knots and quantum algorithms.
We report a deterministic and exact protocol to reverse any unknown qubit-unitary operation, which simulates the time inversion of a closed qubit system. To avoid known no-go results on universal deterministic exact unitary inversion, we consider the most general class of protocols transforming unknown unitary operations within the quantum circuit model, where the input unitary operation is called multiple times in sequence and fixed quantum circuits are inserted between the calls. In the proposed protocol, the input qubit-unitary operation is called 4 times to achieve the inverse operation, and the output state in an auxiliary system can be reused as a catalyst state in another run of the unitary inversion. We also present the simplification of the semidefinite programming for searching the optimal deterministic unitary inversion protocol for an arbitrary dimension presented by M. T. Quintino and D. Ebler [Quantum 6, 679 (2022)2521-327X10.22331/q-2022-03-31-679]. We show a method to reduce the large search space representing all possible protocols, which provides a useful tool for analyzing higher-order quantum transformations for unitary operations.
Matrix representations of quantum operators are computationally complete but often obscure the structural topology of information flow within a quantum circuit \cite{nielsen2000}. In this paper, we introduce a generalized graph-theoretic framework for analyzing quantum operators by mapping unitary matrices to directed graphs; we term these structures \emph{Imperfect Graphs} or more formally as \emph{Topological Structure of Superpositions}(TSS) as a tool to devise better Quantum Algorithms. In this framework, we represent computational basis states as vertices. A directed edge exists between two vertices if and only if there is a non-zero amplitude transition between them, effectively mapping the support of the unitary operator. In this paper we deliberately discard probability amplitudes and phase information to isolate the connectivity and reachability properties of the operator. We demonstrate how TSS intuitively helps describe gates such as the Hadamard, Pauli-(X,Y,Z) gates, etc \cite{nielsen2000}. This framework provides a novel perspective for viewing quantum circuits as discrete dynamical systems \cite{childs2009,aharonov2001}
Keywords: Quantum Algorithms, Unitary Matrix Approach, Topological Structure of Superpositions (TSS), Graph Theory
We give three finite presentations in terms of generators and relations for three groups of operators. The operators are frequently used in quantum computation. The result can potentially be used to optimize quantum circuits. Two of the three proofs use the Reidemeister-Schreier theorem. Both the Reidemeister-Schreier theorem and the two proofs using it have been formally verified in the proof assistant Agda. First, we give a finite presentation of the group of 2-qubit Clifford+T operators. The proof applies the Reidemeister-Schreier theorem to a finite presentation of a supergroup of the 2-qubit Clifford+T group that was given by Greylyn. The process generates hundreds of relations. We then simplify them to 20 relations using a Pauli rotation representation as an almost-normal form. Second, we give a finite presentation of U_n (Z[1/2, i]), the group of unitary n × n- matrices with entries in Z[1/2, i]. Amy, Glaudell, and Ross showed that this group is generated by a certain set of two-level matrices, and they also gave an exact synthesis algorithm that converts any element of U_n (Z[1/2, i]) to a word in these generators. We say that a word is in normal form if it is an output of the exact synthesis algorithm. Using what we call semantically guided rewriting, we show that each word in the two-level generators is equivalent to a normal form under the congruence generated by finitely many relations. Third, we give a finite presentation of the group of 3-qubit Clifford+CS operators. Amy, Glaudell, and Ross proved that up to a condition on the determinant, the n-qubit Clifford+CS group is exactly U_(2^n) (Z[1/2, i]). In particular, the 3-qubit Clifford+CS group is a subgroup of U_8 (Z[1/2, i]) of index 2. Using a similar technique as in our first result, we repeatedly apply the Reidemeister-Schreier theorem to the presentation of U_8 (Z[1/2, i]) (from our second result), and then simplify thousands of relations into 17 relations. To do the simplification, we devise an almost-normal form for Clifford+CS operators. We also show that the 3-qubit Clifford+CS group, which is of course infinite, is the amalgamated product of three finite subgroups.
Scattering processes in high-energy physics are inherently quantum mechanical, yet are typically analyzed at the level of final states, where entanglement appears as a property of the outcome rather than a consequence of the underlying dynamics. We reformulate scattering at the level of the process itself by representing helicity transition matrices as quantum circuits. Once the kinematic configuration and scattering channel are fixed, the problem reduces to a finite-dimensional quantum map, making a circuit description natural. Within this framework, an example of the process e+e−→μ+μ− is shown, which decomposes into unitary and non-unitary components, corresponding to coherent mixing and postselection effects. This representation reorganizes the amplitude into distinct operational elements, providing a perspective in which collider processes can be viewed as constrained quantum circuits and their entanglement structure can be understood in terms of the underlying circuit dynamics, opening the door to analyzing their properties using the language of quantum information.