{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T00:55:02Z","timestamp":1783040102073,"version":"3.54.6"},"reference-count":64,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2024,12,9]],"date-time":"2024-12-09T00:00:00Z","timestamp":1733702400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"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, c<\/mml:mi>o<\/mml:mi>n<\/mml:mi>s<\/mml:mi>t<\/mml:mi>a<\/mml:mi>n<\/mml:mi>t<\/mml:mi><\/mml:math>-depth, quantum circuits that perform 1 qubit gates and n<\/mml:mi>e<\/mml:mi>a<\/mml:mi>r<\/mml:mi>e<\/mml:mi>s<\/mml:mi>t<\/mml:mi>&#x2212;<\/mml:mo>n<\/mml:mi>e<\/mml:mi>i<\/mml:mi>g<\/mml:mi>h<\/mml:mi>b<\/mml:mi>o<\/mml:mi>r<\/mml:mi><\/mml:math> 2 qubit gates.We introduce a new computational model called Local Alternating Quantum Classical Computations<\/mml:mtext><\/mml:mrow><\/mml:math>(LAQCC)<\/mml:mtext><\/mml:mrow><\/mml:math>. 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 LAQCC<\/mml:mtext><\/mml:mrow><\/mml:math> circuit, and that any LAQCC<\/mml:mtext><\/mml:mrow><\/mml:math> circuit can be simulated by a Q<\/mml:mi>N<\/mml:mi>C<\/mml:mi>1<\/mml:mn><\/mml:msup><\/mml:mrow><\/mml:math>circuit. Next, we conjecture the non-simulatability of LAQCC<\/mml:mtext><\/mml:mrow><\/mml:math> by showing that LAQCC<\/mml:mtext><\/mml:mrow><\/mml:math> contains the class of Instantaneous Quantum Polynomial-time circuits. We also show that any LAQCC<\/mml:mtext><\/mml:mrow><\/mml:math> 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 LAQCC<\/mml:mtext><\/mml:mrow><\/mml:math> 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<\/mml:mi><\/mml:math>-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<\/mml:mi><\/mml:math>-bit strings of Hamming weight k<\/mml:mi><\/mml:math>, for k<\/mml:mi>=<\/mml:mo>O<\/mml:mi><\/mml:mrow>(<\/mml:mo>n<\/mml:mi><\/mml:msqrt>)<\/mml:mo><\/mml:math>, but fails for higher k<\/mml:mi><\/mml:math> 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<\/mml:mi><\/mml:math> requiring log-depth circuits that maps between the factoradic number system and the combinatorial number system.<\/jats:p>","DOI":"10.22331\/q-2024-12-09-1552","type":"journal-article","created":{"date-parts":[[2024,12,9]],"date-time":"2024-12-09T17:07:41Z","timestamp":1733764061000},"page":"1552","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":34,"title":["State preparation by shallow circuits using feed forward"],"prefix":"10.22331","volume":"8","author":[{"given":"Harry","family":"Buhrman","sequence":"first","affiliation":[{"name":"QuSoft, CWI & University of Amsterdam, Amsterdam, the Netherlands"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marten","family":"Folkertsma","sequence":"additional","affiliation":[{"name":"QuSoft, CWI & University of Amsterdam, Amsterdam, the Netherlands"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bruno","family":"Loff","sequence":"additional","affiliation":[{"name":"LASIGE & Department of Mathematics, University of Lisbon"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Niels M. 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