State preparation by shallow circuits using feed forward

Harry Buhrman; Marten Folkertsma; Bruno Loff; Niels M. P. Neumann

doi:10.22331/q-2024-12-09-1552

State preparation by shallow circuits using feed forward

Harry Buhrman¹, Marten Folkertsma¹, Bruno Loff², and Niels M. P. Neumann^1,3

¹QuSoft, CWI & University of Amsterdam, Amsterdam, the Netherlands
²LASIGE & Department of Mathematics, University of Lisbon
³The Netherlands Organisation for Applied Scientific Research (TNO), Delft, the Netherlands

Published:	2024-12-09, volume 8, page 1552
Eprint:	arXiv:2307.14840v5
Doi:	https://doi.org/10.22331/q-2024-12-09-1552
Citation:	Quantum 8, 1552 (2024).

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Abstract

Fault tolerant quantum computers repetitively apply a four-step procedure: First, perform a few one and two-qubit quantum gates. Second, perform a syndrome measurement on a subset of the qubits. Third, perform fast classical computations to establish if and where errors occurred. And, fourth, correct the errors with a correction step. The next iteration applies the same procedure with new one and two-qubit gates. Even though current error-rates prohibit this procedure to work and fault tolerant quantum computing remains a distant goal, the same procedure can already prove useful today. In this work we make use of this four-step scheme not to carry out fault-tolerant computations, but to enhance short, $constant$-depth, quantum circuits that perform 1 qubit gates and $nearest-neighbor$ 2 qubit gates.

We introduce a new computational model called $\textit{Local Alternating Quantum Classical Computations}$ $\textsf{(LAQCC)}$. In this model, qubits are placed in a grid and they can only interact with their direct neighbors; the quantum circuits are of constant depth with intermediate measurements; a classical controller can perform log-depth computations on these intermediate measurement outcomes and control future quantum operations based on the outcome. This model fits naturally between quantum algorithms in the NISQ era and full-fledged fault-tolerant quantum computation. We first prove that any Clifford circuit has an equivalent $\textsf{LAQCC}$ circuit, and that any $\textsf{LAQCC}$ circuit can be simulated by a $\mathsf{QNC^1}$circuit. Next, we conjecture the non-simulatability of $\textsf{LAQCC}$ by showing that $\textsf{LAQCC}$ contains the class of Instantaneous Quantum Polynomial-time circuits. We also show that any $\textsf{LAQCC}$ circuit with polynomial-sized quantum circuits and unbounded classical computations is contained in the class of quantum circuits equipped with post-selection gates with respect to the task of state preparation. We continue by presenting $\textsf{LAQCC}$ implementations of different subroutines, including OR-gates, quantum Fourier transforms and Threshold gates. These subroutines prove vital in constructing three state preparation routines in the main part of this work. Preparing a uniform superposition uses constant-depth arithmetic gates, combined with an exact Grover implementation by Long. For the $W$-state, we employ a compress-uncompress method to switch between a binary and one-hot encoding. This method extends to the more generalized Dicke-states, the superposition of $n$-bit strings of Hamming weight $k$, for $k=\mathcal{O}(\sqrt{n})$, but fails for higher $k$ due to the birthday paradox. We extend this protocol to a protocol that prepares many-body scar states, highly excited states with low entanglement and longer coherence times than states with the same energy density. We present a circuit for preparing Dicke-states for larger $k$ requiring log-depth circuits that maps between the factoradic number system and the combinatorial number system.

Featured image: The quantum circuit to prepare the 3-qubit GHZ state using an LAQCC circuit. Double lines indicate classical information and dotted lines the quantum state at various points. This circuit is considered to be of constant depth for size GHZ state. When using this type of circuit for preparing an GHZ state on n qubits, the circuit looks the same, except it is repeated in the width, not increasing the depth.

► BibTeX data

@article{Buhrman2024statepreparation,
  doi = {10.22331/q-2024-12-09-1552},
  url = {https://doi.org/10.22331/q-2024-12-09-1552},
  title = {State preparation by shallow circuits using feed forward},
  author = {Buhrman, Harry and Folkertsma, Marten and Loff, Bruno and Neumann, Niels M. P.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1552},
  month = dec,
  year = {2024}
}

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