On Beam Splittings and Mathematical Construction of Quantum Logical Gate (量子情報と量子カオスの数理)

Wolfgang Freudenberg, Ohya, Masanori, Noboru Watanabe

Keita Kohira, Noboru Watanabe

Sumathi Pawar, Suchetha N V, Ramasubramanian Murugesan +2 more

Zimu Li, Yuguo Shao, Fuchuan Wei +2 more

We establish a systematic mathematical framework for constructing and computing (co)homological invariants that induce logical gates on quantum low-density parity-check (qLDPC) codes. By synthesizing tools from graph theory, algebraic topology, and number theory, we study constructions ranging from hypergraph product (HGP) codes to sheaf codes. Using commutative algebra, we generalize the notion of canonical logical representatives from HGP codes to the sheaf code setting, resolving a longstanding challenge in explicitly characterizing sheaf codewords. Building on this foundation, we present the first comprehensive computation of cup products within the intricate framework of sheaf codes. Given Artin's primitive root conjecture which holds under the generalized Riemann hypothesis, {we establish a nearly linear family of independent cup product cochains on almost-good qLDPC codes. Under a separate nonvanishing pairing condition, they are expected to yield parallel logical multi-controlled-Z gates. Moreover, by interpreting sheaf codes as covering spaces of HGP codes, we construct interlaced families through odd-degree lifts that preserve compatible invariant pairings, allowing nontrivial, constant-depth logical gates throughout an infinite family to be certified on a constant-size HGP seed. Treating local code constructions and expansion bounds as external inputs, our framework identifies the compatibility conditions for combining large dimension and distance, local testability, and constant-depth non-Clifford gates, thereby provides a mathematical foundation for separate constructions that realize these properties simultaneously.

Wolfgang Freudenberg, Masanori Ohya, NATALIYA TURCHINA +1 more

Wolfgang Freudenberg, Masanori Ohya, Noboru Watanabe

In usual computer, there is an upper bound of computational speed because of irreversibility of logical gate.In order to avoid this demerit, Fredkin and Tof- foli [3] proposed a conservative logical gate.After that, Milburn [4] constructed a physical model of reversible quantum gate with beam splittings and a Kerr medium.This model is called FTM (Fredkin $-$ Toffoli $-$ Milburn gate) in this paper.This FTM gate was described by the quantum channel and the efficiency of information transmission of the FTM gate was discussed in [10].FTM gate is using a photon number state as an input state for control gate.The photon number state might be difficult to realize physically.In this paper, we introduced a new device on symmetric Fock space in order to avoid this difficulty.In Section 1, we briefly review noisy quantum channels and generalized beam splittings.In Section 2, we explain the quantum channel for FTM gate based on the notation of Section 1.In Section 3, we intruduced a new operator on symmetric Fock space and discuss the truth table for our new gate.