Measurement-based quantum computation from Clifford quantum cellular automata

TL;DR

The paper addresses how MBQC can be recast as a CAQC based on Clifford QCAs, providing a unified framework that maps MBQC measurements to local rotations within a CAQC block. By leveraging the stabilizer formalism in the Heisenberg picture, the authors derive unit-cell MBQC constructions, demonstrate that simple and entangling CQCAs yield universal CAQC and MBQC, and show how MBQC resource states arise as stabilizer states with local, graph-state-like structure for many cases. They also extend the framework to non-simple CQCAs through alternating-CQCA schemes, proving universality and offering decorated or nonstandard resource states. Finally, the work leverages MBQC-derived PQCs to build hardware-efficient, problem-specific Ansätze, illustrating that different CQCA-based PQCs perform differently on learning tasks and suggesting practical implementations on translationally invariant hardware such as neutral atoms. The results connect MBQC to broader concepts like symmetry-protected topological phases and computational phases of matter, while highlighting a path toward scalable, architecture-friendly quantum computation and learning models.

Abstract

Measurement-based quantum computation (MBQC) is a paradigm for quantum computation where computation is driven by local measurements on a suitably entangled resource state. In this work we show that MBQC is related to a model of quantum computation based on Clifford quantum cellular automata (CQCA). Specifically, we show that certain MBQCs can be directly constructed from CQCAs which yields a simple and intuitive circuit model representation of MBQC in terms of quantum computation based on CQCA. We apply this description to construct various MBQC-based Ansätze for parameterized quantum circuits, demonstrating that the different Ansätze may lead to significantly different performances on different learning tasks. In this way, MBQC yields a family of Hardware-efficient Ansätze that may be adapted to specific problem settings and is particularly well suited for architectures with translationally invariant gates such as neutral atoms.

Figures (13)

• Figure 1: Measurement-based quantum computation (MBQC) as a quantum computation based on Clifford quantum cellular automata (CQCAs). In this work, we will discuss the relation between MBQC (left) where measurements in the $XY$-plane are performed on a suitably entangled resource state and a model of quantum computation based on CQCA (right) where repeated applications of CQCA are interspersed with local $Z$-rotations. This relationship becomes apparent in the Heisenberg picture using the stabilizer formalism.
• Figure 2: Example CQCAs and their action on the basis of observables. CQCAs are translationally invariant, locality preserving unitaries acting on a ring of $N$ qubits that map products of Pauli operators to products of Pauli operators. They are defined by their action on $X_i,Z_i$ for any one qubit $i\in\{1,...,N\}$. (a) Example of the cluster CQCA $T_c$ which is a so-called glider CQCA. (b) Example of a periodic CQCA $\tilde{T}_c$ that has a fixed period $L=2$ for any $N$. (c) Example of a fractal CQCA $\hat{T}_c$ that is neither a glider nor a periodic CQCA. Here $H$ is a Hadamard gate, lines connecting qubits represent $CZ$-gates and $\sqrt{X}=HSH$ where $S=\sqrt{Z}$ is a phase gate.
• Figure 3: Cellular automaton-based quantum computation (CAQC) is constructed from blocks. The block depth of a CAQC is defined by the period $L$ of the corresponding CQCA $T$ such that each block acts as $T^L=id$ if no rotations are present. Each block has $L\times N$ independent parameters, e.g., $\{\theta^j_i\}_{i,j}$ for $j=1,...,L$, $i=1,...,N$ in the first block, defining the rotation angles for $\exp(i\theta^j_i Z_i)$ acting on qubit $i$ at depth $j$. As we will see, each block yields a universal set of gates, parameterized by $\{\theta^j_i\}_{i,j}$, such that their concatenation enables universal quantum computation.
• Figure 4: Constructing a universal gate set for one block of CAQC based on $T_c$. As we consider periodic boundary conditions and CQCAs are translationally invariant, it is sufficient to show that within a constant-size region of qubits (here: qubits $i-1,i,i+1$), we can construct any single-qubit rotation and one entangling gate. Depicted in blue on qubit $i-1$, we show that a $Z$-rotation before the full block is equivalent to a $Z$-rotation at the end of the block as $T_c^L=id$. Depicted in orange on qubit $i$ we show that a $Z$-rotation at depth $L-1$ within the block (where we count depth by the layers of $Z$-rotations), yields a 3-qubit entangling gate as $T_c^2(Z_i)=X_{i-1}Z_iX_{i+1}$. Depicted in magenta on qubit $i+1$ we show that a $Z$-rotation at depth $L$ within the block, yields a $X$-rotation as $T(Z_{i+1})=X_{i+1}$.
• Figure 5: Example of a graph state stabilizer. Shown are the six canonical generators of the stabilizer group (as defined by Eq. \ref{['eq:graph_stab']}) for one specific 6-qubit graph state. They are not unique and can be multiplied in a way that yields six different generators.

Theorems & Definitions (8)

• Lemma 1
• Theorem 2
• Theorem 3
• Corollary 4
• Lemma 5
• Theorem 6
• Proposition 7
• Proposition 8