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**Published in:** LIPIcs, Volume 266, 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)

We define rewinding operators that invert quantum measurements. Then, we define complexity classes RwBQP, CBQP, and AdPostBQP as sets of decision problems solvable by polynomial-size quantum circuits with a polynomial number of rewinding operators, cloning operators, and adaptive postselections, respectively. Our main result is that BPP^PP ⊆ RwBQP = CBQP = AdPostBQP ⊆ PSPACE. As a byproduct of this result, we show that any problem in PostBQP can be solved with only postselections of outputs whose probabilities are polynomially close to one. Under the strongly believed assumption that BQP ⊉ SZK, or the shortest independent vectors problem cannot be efficiently solved with quantum computers, we also show that a single rewinding operator is sufficient to achieve tasks that are intractable for quantum computation. In addition, we consider rewindable Clifford and instantaneous quantum polynomial time circuits.

Ryo Hiromasa, Akihiro Mizutani, Yuki Takeuchi, and Seiichiro Tani. Rewindable Quantum Computation and Its Equivalence to Cloning and Adaptive Postselection. In 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 266, pp. 9:1-9:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{hiromasa_et_al:LIPIcs.TQC.2023.9, author = {Hiromasa, Ryo and Mizutani, Akihiro and Takeuchi, Yuki and Tani, Seiichiro}, title = {{Rewindable Quantum Computation and Its Equivalence to Cloning and Adaptive Postselection}}, booktitle = {18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)}, pages = {9:1--9:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-283-9}, ISSN = {1868-8969}, year = {2023}, volume = {266}, editor = {Fawzi, Omar and Walter, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2023.9}, URN = {urn:nbn:de:0030-drops-183193}, doi = {10.4230/LIPIcs.TQC.2023.9}, annote = {Keywords: Quantum computing, Postselection, Lattice problems} }

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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Space-bounded computation has been a central topic in classical and quantum complexity theory. In the quantum case, every elementary gate must be unitary. This restriction makes it unclear whether the power of space-bounded computation changes by allowing intermediate measurement. In the bounded error case, Fefferman and Remscrim [STOC 2021, pp.1343-1356] and Girish, Raz and Zhan [ICALP 2021, pp.73:1-73:20] recently provided the break-through results that the power does not change. This paper shows that a similar result holds for space-bounded quantum computation with postselection. Namely, it is proved possible to eliminate intermediate postselections and measurements in the space-bounded quantum computation in the bounded-error setting. Our result strengthens the recent result by Le Gall, Nishimura and Yakaryilmaz [TQC 2021, pp.10:1-10:17] that logarithmic-space bounded-error quantum computation with intermediate postselections and measurements is equivalent in computational power to logarithmic-space unbounded-error probabilistic computation. As an application, it is shown that bounded-error space-bounded one-clean qubit computation (DQC1) with postselection is equivalent in computational power to unbounded-error space-bounded probabilistic computation, and the computational supremacy of the bounded-error space-bounded DQC1 is interpreted in complexity-theoretic terms.

Seiichiro Tani. Space-Bounded Unitary Quantum Computation with Postselection. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 81:1-81:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{tani:LIPIcs.MFCS.2022.81, author = {Tani, Seiichiro}, title = {{Space-Bounded Unitary Quantum Computation with Postselection}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {81:1--81:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.81}, URN = {urn:nbn:de:0030-drops-168798}, doi = {10.4230/LIPIcs.MFCS.2022.81}, annote = {Keywords: quantum complexity theory, space-bounded computation, postselection} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

We study the effect of noise on the classical simulatability of quantum circuits defined by computationally tractable (CT) states and efficiently computable sparse (ECS) operations. Examples of such circuits, which we call CT-ECS circuits, are IQP, Clifford Magic, and conjugated Clifford circuits. This means that there exist various CT-ECS circuits such that their output probability distributions are anti-concentrated and not classically simulatable in the noise-free setting (under plausible assumptions). First, we consider a noise model where a depolarizing channel with an arbitrarily small constant rate is applied to each qubit at the end of computation. We show that, under this noise model, if an approximate value of the noise rate is known, any CT-ECS circuit with an anti-concentrated output probability distribution is classically simulatable. This indicates that the presence of small noise drastically affects the classical simulatability of CT-ECS circuits. Then, we consider an extension of the noise model where the noise rate can vary with each qubit, and provide a similar sufficient condition for classically simulating CT-ECS circuits with anti-concentrated output probability distributions.

Yasuhiro Takahashi, Yuki Takeuchi, and Seiichiro Tani. Classically Simulating Quantum Circuits with Local Depolarizing Noise. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 83:1-83:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{takahashi_et_al:LIPIcs.MFCS.2020.83, author = {Takahashi, Yasuhiro and Takeuchi, Yuki and Tani, Seiichiro}, title = {{Classically Simulating Quantum Circuits with Local Depolarizing Noise}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {83:1--83:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.83}, URN = {urn:nbn:de:0030-drops-127533}, doi = {10.4230/LIPIcs.MFCS.2020.83}, annote = {Keywords: Classical Simulation, Quantum Circuit, Local Depolarizing Noise} }

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**Published in:** LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

An ordered binary decision diagram (OBDD) is a directed acyclic graph that represents a Boolean function. Since OBDDs have many nice properties as data structures, they have been extensively studied for decades in both theoretical and practical fields, such as VLSI (Very Large Scale Integration) design, formal verification, machine learning, and combinatorial problems. Arguably, the most crucial problem in using OBDDs is that they may vary exponentially in size depending on their variable ordering (i.e., the order in which the variables are to be read) when they represent the same function. Indeed, it is NP hard to find an optimal variable ordering that minimizes an OBDD for a given function. Friedman and Supowit provided a clever deterministic algorithm with time/space complexity O^∗(3ⁿ), where n is the number of variables of the function, which is much better than the trivial brute-force bound O^∗(n!2ⁿ). This paper shows that a further speedup is possible with quantum computers by presenting a quantum algorithm that produces a minimum OBDD together with the corresponding variable ordering in O^∗(2.77286ⁿ) time and space with an exponentially small error probability. Moreover, this algorithm can be adapted to constructing other minimum decision diagrams such as zero-suppressed BDDs.

Seiichiro Tani. Quantum Algorithm for Finding the Optimal Variable Ordering for Binary Decision Diagrams. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 36:1-36:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{tani:LIPIcs.SWAT.2020.36, author = {Tani, Seiichiro}, title = {{Quantum Algorithm for Finding the Optimal Variable Ordering for Binary Decision Diagrams}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {36:1--36:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-150-4}, ISSN = {1868-8969}, year = {2020}, volume = {162}, editor = {Albers, Susanne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.36}, URN = {urn:nbn:de:0030-drops-122832}, doi = {10.4230/LIPIcs.SWAT.2020.36}, annote = {Keywords: Binary Decision Diagram, Variable Ordering, Quantum Algorithm} }

Document

**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

We study the computational power of shallow quantum circuits
with O(log n) initialized and n^{O(1)} uninitialized ancillary
qubits, where n is the input length and the initial state of
the uninitialized ancillary qubits is arbitrary. First, we show
that such a circuit can compute any symmetric function on n bits
that is classically computable in polynomial time. Then, we
regard such a circuit as an oracle and show that a
polynomial-time classical algorithm with the oracle can estimate
the elements of any unitary matrix corresponding to a
constant-depth quantum circuit on n qubits. Since it seems unlikely
that these tasks can be done with only O(log n) initialized
ancillary qubits, our results give evidences that adding
uninitialized ancillary qubits increases the computational power
of shallow quantum circuits with only O(log n) initialized
ancillary qubits. Lastly, to understand the limitations of
uninitialized ancillary qubits, we focus on
near-logarithmic-depth quantum circuits with them and show
the impossibility of computing the parity function on n bits.

Yasuhiro Takahashi and Seiichiro Tani. Power of Uninitialized Qubits in Shallow Quantum Circuits. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 57:1-57:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{takahashi_et_al:LIPIcs.STACS.2018.57, author = {Takahashi, Yasuhiro and Tani, Seiichiro}, title = {{Power of Uninitialized Qubits in Shallow Quantum Circuits}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {57:1--57:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.57}, URN = {urn:nbn:de:0030-drops-84907}, doi = {10.4230/LIPIcs.STACS.2018.57}, annote = {Keywords: quantum circuit complexity, shallow quantum circuit, uninitialized qubit} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

This paper investigates the power of polynomial-time quantum computation in which only a very limited number of qubits are initially clean in the |0> state, and all the remaining qubits are initially in the totally mixed state. No initializations of qubits are allowed during the computation, nor are intermediate measurements. The main contribution of this paper is to develop unexpectedly strong error-reduction methods for such quantum computations that simultaneously reduce the number of necessary clean qubits. It is proved that any problem solvable by a polynomialtime quantum computation with one-sided bounded error that uses logarithmically many clean qubits is also solvable with exponentially small one-sided error using just two clean qubits, and with polynomially small one-sided error using just one clean qubit. It is further proved in the twosided-error case that any problem solvable by such a computation with a constant gap between completeness and soundness using logarithmically many clean qubits is also solvable with exponentially small two-sided error using just two clean qubits. If only one clean qubit is available, the problem is again still solvable with exponentially small error in one of the completeness and soundness and with polynomially small error in the other. An immediate consequence is that the Trace Estimation problem defined with fixed constant threshold parameters is complete for BQ_{[1]}P and BQ_{log}P, the classes of problems solvable by polynomial-time quantum computations with completeness 2/3 and soundness 1/3 using just one and logarithmically many clean qubits, respectively. The techniques used for proving the error-reduction results may be of independent interest in themselves, and one of the technical tools can also be used to show the hardness of weak classical simulations of one-clean-qubit computations (i.e., DQC1 computations).

Keisuke Fujii, Hirotada Kobayashi, Tomoyuki Morimae, Harumichi Nishimura, Shuhei Tamate, and Seiichiro Tani. Power of Quantum Computation with Few Clean Qubits. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fujii_et_al:LIPIcs.ICALP.2016.13, author = {Fujii, Keisuke and Kobayashi, Hirotada and Morimae, Tomoyuki and Nishimura, Harumichi and Tamate, Shuhei and Tani, Seiichiro}, title = {{Power of Quantum Computation with Few Clean Qubits}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {13:1--13:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.13}, URN = {urn:nbn:de:0030-drops-62960}, doi = {10.4230/LIPIcs.ICALP.2016.13}, annote = {Keywords: DQC1, quantum computing, complete problems, error reduction} }

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