Quantum Convolution of Linearly Recursive Sequences

Chen Yang, Kodai Kanemaru, Norio Yoshida +2 more

Discrete circular convolution over $\mathbb{Z}/N\mathbb{Z}$ is a linear operator and can be implemented on quantum hardware within the linear-combination-of-unitaries (LCU) framework. In this work, we make this connection explicit through an asymmetric-LCU formulation: circular convolution is the postselected block of a circuit whose controlled-shift unitary is modular addition on computational-basis states. The asymmetry is essential: fixing the postselection state to the uniform state $|u\rangle$ while supplying the kernel state $|\mathbf{b}\rangle$ as the input ancilla naturally preserves the complex coefficients $b_i$ within the block, whereas a symmetric overlap would yield $|b_i|^2$ weights and erase their phases. Accordingly, when $|\mathbf{a}\rangle$ and $|\mathbf{b}\rangle$ are supplied by upstream quantum routines, the convolution subroutine requires only the fixed uncompute $\mathrm{PREP}_u^\dagger$, completely avoiding the need for a kernel-dependent inverse preparation $\mathrm{PREP}_b^\dagger$. We then introduce a reversal matrix $J_n=X^{\otimes n}$ and define reflected shifts $\widetilde{L}_{i,n}=L_{i,n}J_n$. This symmetrization yields a recursive operator algebra for convolution that is natively compatible with LCU/block-encoding workflows. The resulting symmetrized operator differs from circular convolution only by one known input-side $J_n$ layer. Crucially, for real-valued kernels, the resulting operator $H_n(\mathbf{b})=\sum_i b_i\widetilde{L}_{i,n}$ is Hermitian, providing a direct Hermitian interface for quantum singular value transformation (QSVT) and related spectral transformations. Based on this framework, we present a transparent recursive construction, paired with an exactly equivalent optimized bitwise compilation of the same $\mathrm{SELECT}$ block. Finally, we evaluate implementation trade-offs and resource scaling under explicit cost-model conventions.

Vladimir Vargas-Calderón

Masanori Ohya, Igor Vasil'evich Volovich

Monireh Houshmand, Mark M. Wilde

Poulin discovered an important separation between the classical and quantum theories of convolutional coding, by proving that a quantum convolutional encoder cannot be both noncatastrophic and recursive. Noncatastrophicity is desirable so that an iterative decoding algorithm converges when decoding a quantum turbo code whose constituents are quantum convolutional codes, and recursiveness is as well so that a quantum turbo code has a minimum distance growing nearly linearly with the length of the code, respectively. Their proof of the aforementioned theorem was admittedly “rather involved,” and as such, it has been desirable since their result to find a simpler proof. In this paper, we furnish a proof that is arguably simpler. Our approach is group-theoretic-we show that the subgroup of memory states that are part of a zero physical-weight cycle of a quantum convolutional encoder is equivalent to the centralizer of its “finite-memory” subgroup (the subgroup of memory states which eventually reach the identity memory state by identity operator inputs for the information qubits and identity or Pauli- Z operator inputs for the ancilla qubits). After proving that this symmetry holds for any quantum convolutional encoder, it easily follows that an encoder is nonrecursive if it is noncatastrophic. Our proof also illuminates why this no-go theorem does not apply to entanglement-assisted quantum convolutional encoders-the introduction of shared entanglement as a resource allows the above symmetry to be broken.

Rihab Hoceini, Ahmed Bouida

Monireh Houshmand, Mark M. Wilde

Poulin discovered an important separation between the classical and quantum theories of convolutional coding, by proving that a quantum convolutional encoder cannot be both noncatastrophic and recursive. Noncatastrophicity is desirable so that an iterative decoding algorithm converges when decoding a quantum turbo code whose constituents are quantum convolutional codes, and recursiveness is as well so that a quantum turbo code has a minimum distance growing nearly linearly with the length of the code, respectively. Their proof of the aforementioned theorem was admittedly 'rather involved,' and as such, it has been desirable since their result to find a simpler proof. In this paper, we furnish a proof that is arguably simpler. Our approach is group-theoretic-we show that the subgroup of memory states that are part of a zero physical-weight cycle of a quantum convolutional encoder is equivalent to the centralizer of its 'finite-memory' subgroup (the subgroup of memory states which eventually reach the identity memory state by identity operator inputs for the information qubits and identity or Pauli-Z operator inputs for the ancilla qubits). After proving that this symmetry holds for any quantum convolutional encoder, it easily follows that an encoder is nonrecursive if it is noncatastrophic. Our proof also illuminates why this no-go theorem does not apply to entanglement-assisted quantum convolutional encoders-the introduction of shared entanglement as a resource allows the above symmetry to be broken. © 1963-2012 IEEE.