Elucidating the Physical and Mathematical Properties of the Prouhet-Thue-Morse Sequence in Quantum Computing

TL;DR

The paper investigates how the Prouhet-Thue-Morse sequence can be embedded in quantum computing architectures and protocols. It defines PTM logical states, relates them to Hadamard transforms, and shows their emergence as eigenstates of XX interactions and as eigenvalues of $S_x$, enabling robust memory encoding. It demonstrates applications to quantum error correction, noise-resistant memories, and links to quantum chaos and number theory, including connections to the Riemann zeta function and Dirichlet series. The work highlights interdisciplinary avenues bridging mathematics and quantum physics and proposes a pathway to more fault-tolerant quantum technologies.

Abstract

This study explores the applications of the Prouhet-Thue-Morse (PTM) sequence in quantum computing, highlighting its mathematical elegance and practical relevance. We demonstrate the critical role of the PTM sequence in quantum error correction, in noise-resistant quantum memories, and in providing insights into quantum chaos. Notably, we demonstrate how the PTM sequence naturally appears in Ising X-X interacting systems, leading to a proposed robust encoding of quantum memories in such systems. Furthermore, connections to number theory, including the Riemann zeta function, bridge quantum computing with pure mathematics. Our findings emphasize the PTM sequence's importance in understanding the mathematical structure of quantum computing systems and the development of the full potential of quantum technologies and invite further interdisciplinary research.

Figures (6)

• Figure 1: Definition of the PTM Gate acting on $N$ qubits.
• Figure 2: Schematic representation of the interplay between the decoherence behavior of a $X$-$X$ Ising chain initialized in the GHZ state, and the PTM sequence.
• Figure 3: Time-evolution of the populations $P$ of different states initialized in $\ket{\psi_+(0)} = \frac{1}{\sqrt{2}}\left(\ket{(0)_2} + \ket{(N)_2}\right)$ (red) or $\ket{\psi_-(0)} = \frac{1}{\sqrt{2}}\left(\ket{(0)_2} - \ket{(N)_2}\right)$ (blue). With $\ket{\psi_\pm(t)} = \text{exp}\left(\frac{-i\pi}{2\sqrt{2}}\left(S_z+S_x\right)t\right)\ket{\psi_\pm(0)}$. The final state corresponds to $\ket{\psi_\pm(1)} = \bigotimes_{k=1}^N H^{(k)}\ket{\psi_\pm(0)}$.
• Figure 4: Traditional (Shor's) 3-qubit phase-flip error correction code. One can detect and correct $1$ phase flip error using 2 ancilla qubits for parity checks, and with initial state $\ket{\phi}=\alpha\ket{000}+\beta\ket{111}$. CORR takes as an input the classical two-bits registry $c=(i)_2$, does nothing if $i=0$, and flips qubit $i$ otherwise, counting from top to bottom.
• Figure 5: 3-qubit single phase-flip error correction code for one logical PTM state.