9 Search Results for "Remscrim, Zachary"

Document
DOI: 10.4230/LIPIcs.TQC.2025.11
Quantum Catalytic Space

Authors: Harry Buhrman, Marten Folkertsma, Ian Mertz, Florian Speelman, Sergii Strelchuk, Sathyawageeswar Subramanian, and Quinten Tupker

Published in: LIPIcs, Volume 350, 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025)

Abstract
Space complexity is a key field of study in theoretical computer science. In the quantum setting there are clear motivations to understand the power of space-restricted computation, as qubits are an especially precious and limited resource. Recently, a new branch of space-bounded complexity called catalytic computing has shown that reusing space is a very powerful computational resource, especially for subroutines that incur little to no space overhead. While quantum catalysis in an information theoretic context, and the power of "dirty" qubits for quantum computation, has been studied over the years, these models are generally not suitable for use in quantum space-bounded algorithms, as they either rely on specific catalytic states or destroy the memory being borrowed. We define the notion of catalytic computing in the quantum setting and show a number of initial results about the model. First, we show that quantum catalytic logspace can always be computed quantumly in polynomial time; the classical analogue of this is the largest open question in catalytic computing. This also allows quantum catalytic space to be defined in an equivalent way with respect to circuits instead of Turing machines. We also prove that quantum catalytic logspace can simulate log-depth threshold circuits, a class which is known to contain (and believed to strictly contain) quantum logspace, thus showcasing the power of quantum catalytic space. Finally we show that both unitary quantum catalytic logspace and classical catalytic logspace can be simulated in the one-clean qubit model.
Cite as

Harry Buhrman, Marten Folkertsma, Ian Mertz, Florian Speelman, Sergii Strelchuk, Sathyawageeswar Subramanian, and Quinten Tupker. Quantum Catalytic Space. In 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 350, pp. 11:1-11:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{buhrman_et_al:LIPIcs.TQC.2025.11,
  author =	{Buhrman, Harry and Folkertsma, Marten and Mertz, Ian and Speelman, Florian and Strelchuk, Sergii and Subramanian, Sathyawageeswar and Tupker, Quinten},
  title =	{{Quantum Catalytic Space}},
  booktitle =	{20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025)},
  pages =	{11:1--11:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-392-8},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{350},
  editor =	{Fefferman, Bill},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2025.11},
  URN =		{urn:nbn:de:0030-drops-240606},
  doi =		{10.4230/LIPIcs.TQC.2025.11},
  annote =	{Keywords: quantum computing, quantum complexity, space-bounded algorithms, catalytic computation, one clean qubit}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.17
Space-Bounded Quantum Interactive Proof Systems

Authors: François Le Gall, Yupan Liu, Harumichi Nishimura, and Qisheng Wang

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
We introduce two models of space-bounded quantum interactive proof systems, QIPL and QIP_{U}L. The QIP_{U}L model, a space-bounded variant of quantum interactive proofs (QIP) introduced by Watrous (CC 2003) and Kitaev and Watrous (STOC 2000), restricts verifier actions to unitary circuits. In contrast, QIPL allows logarithmically many pinching intermediate measurements per verifier action, making it the weakest model that encompasses the classical model of Condon and Ladner (JCSS 1995). We characterize the computational power of QIPL and QIP_{U}L. When the message number m is polynomially bounded, QIP_{U}L ⊊ QIPL unless P = NP: - QIPL^HC, a subclass of QIPL defined by a high-concentration condition on yes instances, exactly characterizes NP. - QIP_{U}L is contained in P and contains SAC¹ ∪ BQL, where SAC¹ denotes problems solvable by classical logarithmic-depth, semi-unbounded fan-in circuits. However, this distinction vanishes when m is constant. Our results further indicate that (pinching) intermediate measurements uniquely impact space-bounded quantum interactive proofs, unlike in space-bounded quantum computation, where BQL = BQ_{U}L. We also introduce space-bounded unitary quantum statistical zero-knowledge (QSZK_{U}L), a specific form of QIP_{U}L proof systems with statistical zero-knowledge against any verifier. This class is a space-bounded variant of quantum statistical zero-knowledge (QSZK) defined by Watrous (SICOMP 2009). We prove that QSZK_{U}L = BQL, implying that the statistical zero-knowledge property negates the computational advantage typically gained from the interaction.
Cite as

François Le Gall, Yupan Liu, Harumichi Nishimura, and Qisheng Wang. Space-Bounded Quantum Interactive Proof Systems. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 17:1-17:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{legall_et_al:LIPIcs.CCC.2025.17,
  author =	{Le Gall, Fran\c{c}ois and Liu, Yupan and Nishimura, Harumichi and Wang, Qisheng},
  title =	{{Space-Bounded Quantum Interactive Proof Systems}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{17:1--17:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.17},
  URN =		{urn:nbn:de:0030-drops-237115},
  doi =		{10.4230/LIPIcs.CCC.2025.17},
  annote =	{Keywords: Intermediate measurements, Quantum interactive proofs, Space-bounded quantum computation}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.18
Directed st-Connectivity with Few Paths Is in Quantum Logspace

Authors: Simon Apers and Roman Edenhofer

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
We present a BQSPACE(O(log n))-procedure to count st-paths on directed graphs for which we are promised that there are at most polynomially many paths starting in s and polynomially many paths ending in t. For comparison, the best known classical upper bound in this case just to decide st-connectivity is DSPACE(O(log² n/ log log n)). The result establishes a new relationship between BQL and unambiguity and fewness subclasses of NL. Further, we also show how to recognize directed graphs with at most polynomially many paths between any two nodes in BQSPACE(O(log n)). This yields the first natural candidate for a language separating BQL from 𝖫 and BPL. Until now, all candidates potentially separating these classes were inherently promise problems.
Cite as

Simon Apers and Roman Edenhofer. Directed st-Connectivity with Few Paths Is in Quantum Logspace. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{apers_et_al:LIPIcs.CCC.2025.18,
  author =	{Apers, Simon and Edenhofer, Roman},
  title =	{{Directed st-Connectivity with Few Paths Is in Quantum Logspace}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{18:1--18:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.18},
  URN =		{urn:nbn:de:0030-drops-237128},
  doi =		{10.4230/LIPIcs.CCC.2025.18},
  annote =	{Keywords: Quantum computation, Space-bounded complexity classes, Graph connectivity, Unambiguous computation, Random walks}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.33
Bosonic Quantum Computational Complexity

Authors: Ulysse Chabaud, Michael Joseph, Saeed Mehraban, and Arsalan Motamedi

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
In recent years, quantum computing involving physical systems with continuous degrees of freedom, such as the bosonic quantum states of light, has attracted significant interest. However, a well-defined quantum complexity theory for these bosonic computations over infinite-dimensional Hilbert spaces is missing. In this work, we lay the foundations for such a research program. We introduce natural complexity classes and problems based on bosonic generalizations of BQP, the local Hamiltonian problem, and QMA. We uncover several relationships and subtle differences between standard Boolean classical and discrete-variable quantum complexity classes, and identify outstanding open problems. Our main contributions include the following: 1) Bosonic computations. We show that the power of Gaussian computations up to logspace reductions is equivalent to bounded-error quantum logspace (BQL, characterized by the problem of inverting well-conditioned matrices). More generally, we define classes of continuous-variable quantum polynomial time computations with a bounded probability of error (CVBQP) based on gates generated by polynomial bosonic Hamiltonians and particle-number measurements. Due to the infinite-dimensional Hilbert space, it is not a priori clear whether a decidable upper bound can be obtained for these classes. We identify complete problems for these classes, and we demonstrate a BQP lower bound and an EXPSPACE upper bound by proving bounds on the average energy throughout the computation. We further show that the problem of computing expectation values of polynomial bosonic observables at the output of bosonic quantum circuits using Gaussian and cubic phase gates is in PSPACE. 2) Bosonic ground energy problems. We prove that the problem of deciding whether the spectrum of a bosonic Hamiltonian is bounded from below is co-NP-hard. Furthermore, we show that the problem of finding the minimum energy of a bosonic Hamiltonian critically depends on the non-Gaussian stellar rank of the family of energy-constrained states one optimizes over: for zero stellar rank, i.e., optimizing over Gaussian states, it is NP-complete; for polynomially-bounded stellar rank, it is in QMA; for unbounded stellar rank, it is RE-hard, i.e., undecidable.
Cite as

Ulysse Chabaud, Michael Joseph, Saeed Mehraban, and Arsalan Motamedi. Bosonic Quantum Computational Complexity. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 33:1-33:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chabaud_et_al:LIPIcs.ITCS.2025.33,
  author =	{Chabaud, Ulysse and Joseph, Michael and Mehraban, Saeed and Motamedi, Arsalan},
  title =	{{Bosonic Quantum Computational Complexity}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{33:1--33:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.33},
  URN =		{urn:nbn:de:0030-drops-226612},
  doi =		{10.4230/LIPIcs.ITCS.2025.33},
  annote =	{Keywords: continuous-variable quantum computing, infinite-dimensional quantum systems, stellar rank, Hamiltonian complexity}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2024.9
Quantum Merlin-Arthur and Proofs Without Relative Phase

Authors: Roozbeh Bassirian, Bill Fefferman, and Kunal Marwaha

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract
We study a variant of QMA where quantum proofs have no relative phase (i.e. non-negative amplitudes, up to a global phase). If only completeness is modified, this class is equal to QMA [Grilo et al., 2014]; but if both completeness and soundness are modified, the class (named QMA+ by Jeronimo and Wu [Jeronimo and Wu, 2023]) can be much more powerful. We show that QMA+ with some constant gap is equal to NEXP, yet QMA+ with some other constant gap is equal to QMA. One interpretation is that Merlin’s ability to "deceive" originates from relative phase at least as much as from entanglement, since QMA(2) ⊆ NEXP.
Cite as

Roozbeh Bassirian, Bill Fefferman, and Kunal Marwaha. Quantum Merlin-Arthur and Proofs Without Relative Phase. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bassirian_et_al:LIPIcs.ITCS.2024.9,
  author =	{Bassirian, Roozbeh and Fefferman, Bill and Marwaha, Kunal},
  title =	{{Quantum Merlin-Arthur and Proofs Without Relative Phase}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{9:1--9:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.9},
  URN =		{urn:nbn:de:0030-drops-195370},
  doi =		{10.4230/LIPIcs.ITCS.2024.9},
  annote =	{Keywords: quantum complexity, QMA(2), PCPs}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.54
The Importance of the Spectral Gap in Estimating Ground-State Energies

Authors: Abhinav Deshpande, Alexey V. Gorshkov, and Bill Fefferman

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
The field of quantum Hamiltonian complexity lies at the intersection of quantum many-body physics and computational complexity theory, with deep implications to both fields. The main object of study is the Local Hamiltonian problem, which is concerned with estimating the ground-state energy of a local Hamiltonian and is complete for the class QMA, a quantum generalization of the class NP. A major challenge in the field is to understand the complexity of the Local Hamiltonian problem in more physically natural parameter regimes. One crucial parameter in understanding the ground space of any Hamiltonian in many-body physics is the spectral gap, which is the difference between the smallest two eigenvalues. Despite its importance in quantum many-body physics, the role played by the spectral gap in the complexity of the Local Hamiltonian problem is less well-understood. In this work, we make progress on this question by considering the precise regime, in which one estimates the ground-state energy to within inverse exponential precision. Computing ground-state energies precisely is a task that is important for quantum chemistry and quantum many-body physics. In the setting of inverse-exponential precision (promise gap), there is a surprising result that the complexity of Local Hamiltonian is magnified from QMA to PSPACE, the class of problems solvable in polynomial space (but possibly exponential time). We clarify the reason behind this boost in complexity. Specifically, we show that the full complexity of the high precision case only comes about when the spectral gap is exponentially small. As a consequence of the proof techniques developed to show our results, we uncover important implications for the representability and circuit complexity of ground states of local Hamiltonians, the theory of uniqueness of quantum witnesses, and techniques for the amplification of quantum witnesses in the presence of postselection.
Cite as

Abhinav Deshpande, Alexey V. Gorshkov, and Bill Fefferman. The Importance of the Spectral Gap in Estimating Ground-State Energies. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 54:1-54:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{deshpande_et_al:LIPIcs.ITCS.2022.54,
  author =	{Deshpande, Abhinav and Gorshkov, Alexey V. and Fefferman, Bill},
  title =	{{The Importance of the Spectral Gap in Estimating Ground-State Energies}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{54:1--54:6},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.54},
  URN =		{urn:nbn:de:0030-drops-156501},
  doi =		{10.4230/LIPIcs.ITCS.2022.54},
  annote =	{Keywords: Local Hamiltonian problem, PSPACE, PP, QMA}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.76
Eliminating Intermediate Measurements Using Pseudorandom Generators

Authors: Uma Girish and Ran Raz

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
We show that quantum algorithms of time T and space S ≥ log T with unitary operations and intermediate measurements can be simulated by quantum algorithms of time T ⋅ poly (S) and space {O}(S⋅ log T) with unitary operations and without intermediate measurements. The best results prior to this work required either Ω(T) space (by the deferred measurement principle) or poly(2^S) time [Bill Fefferman and Zachary Remscrim, 2021; Uma Girish et al., 2021]. Our result is thus a time-efficient and space-efficient simulation of algorithms with unitary operations and intermediate measurements by algorithms with unitary operations and without intermediate measurements. To prove our result, we study pseudorandom generators for quantum space-bounded algorithms. We show that (an instance of) the INW pseudorandom generator for classical space-bounded algorithms [Russell Impagliazzo et al., 1994] also fools quantum space-bounded algorithms. More precisely, we show that for quantum space-bounded algorithms that have access to a read-once tape consisting of random bits, the final state of the algorithm when the random bits are drawn from the uniform distribution is nearly identical to the final state when the random bits are drawn using the INW pseudorandom generator. This result applies to general quantum algorithms which can apply unitary operations, perform intermediate measurements and reset qubits.
Cite as

Uma Girish and Ran Raz. Eliminating Intermediate Measurements Using Pseudorandom Generators. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 76:1-76:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{girish_et_al:LIPIcs.ITCS.2022.76,
  author =	{Girish, Uma and Raz, Ran},
  title =	{{Eliminating Intermediate Measurements Using Pseudorandom Generators}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{76:1--76:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.76},
  URN =		{urn:nbn:de:0030-drops-156726},
  doi =		{10.4230/LIPIcs.ITCS.2022.76},
  annote =	{Keywords: quantum algorithms, intermediate measurements, deferred measurement, pseudorandom generator, INW generator}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.39
Lower Bounds on the Running Time of Two-Way Quantum Finite Automata and Sublogarithmic-Space Quantum Turing Machines

Authors: Zachary Remscrim

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
The two-way finite automaton with quantum and classical states (2QCFA), defined by Ambainis and Watrous, is a model of quantum computation whose quantum part is extremely limited; however, as they showed, 2QCFA are surprisingly powerful: a 2QCFA with only a single-qubit can recognize the language L_{pal} = {w ∈ {a,b}^*:w is a palindrome} with bounded error in expected time 2^{O(n)}. We prove that their result cannot be improved upon: a 2QCFA (of any size) cannot recognize L_{pal} with bounded error in expected time 2^{o(n)}. This is the first example of a language that can be recognized with bounded error by a 2QCFA in exponential time but not in subexponential time. Moreover, we prove that a quantum Turing machine (QTM) running in space o(log n) and expected time 2^{n^{1-Ω(1)}} cannot recognize L_{pal} with bounded error; again, this is the first lower bound of its kind. Far more generally, we establish a lower bound on the running time of any 2QCFA or o(log n)-space QTM that recognizes any language L in terms of a natural "hardness measure" of L. This allows us to exhibit a large family of languages for which we have asymptotically matching lower and upper bounds on the running time of any such 2QCFA or QTM recognizer.
Cite as

Zachary Remscrim. Lower Bounds on the Running Time of Two-Way Quantum Finite Automata and Sublogarithmic-Space Quantum Turing Machines. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 39:1-39:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{remscrim:LIPIcs.ITCS.2021.39,
  author =	{Remscrim, Zachary},
  title =	{{Lower Bounds on the Running Time of Two-Way Quantum Finite Automata and Sublogarithmic-Space Quantum Turing Machines}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{39:1--39:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.39},
  URN =		{urn:nbn:de:0030-drops-135781},
  doi =		{10.4230/LIPIcs.ITCS.2021.39},
  annote =	{Keywords: Quantum computation, Lower bounds, Finite automata}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2020.139
The Power of a Single Qubit: Two-Way Quantum Finite Automata and the Word Problem

Authors: Zachary Remscrim

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract
The two-way finite automaton with quantum and classical states (2QCFA), defined by Ambainis and Watrous, is a model of quantum computation whose quantum part is extremely limited; however, as they showed, 2QCFA are surprisingly powerful: a 2QCFA, with a single qubit, can recognize, with bounded error, the language L_{eq} = {a^m b^m :m ∈ ℕ} in expected polynomial time and the language L_{pal} = {w ∈ {a,b}^*:w is a palindrome} in expected exponential time. We further demonstrate the power of 2QCFA by showing that they can recognize the word problems of many groups. In particular 2QCFA, with a single qubit and algebraic number transition amplitudes, can recognize, with bounded error, the word problem of any finitely generated virtually abelian group in expected polynomial time, as well as the word problems of a large class of linear groups in expected exponential time. This latter class (properly) includes all groups with context-free word problem. We also exhibit results for 2QCFA with any constant number of qubits. As a corollary, we obtain a direct improvement on the original Ambainis and Watrous result by showing that L_{eq} can be recognized by a 2QCFA with better parameters. As a further corollary, we show that 2QCFA can recognize certain non-context-free languages in expected polynomial time. In a companion paper, we prove matching lower bounds, thereby showing that the class of languages recognizable with bounded error by a 2QCFA in expected subexponential time is properly contained in the class of languages recognizable with bounded error by a 2QCFA in expected exponential time.
Cite as

Zachary Remscrim. The Power of a Single Qubit: Two-Way Quantum Finite Automata and the Word Problem. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 139:1-139:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{remscrim:LIPIcs.ICALP.2020.139,
  author =	{Remscrim, Zachary},
  title =	{{The Power of a Single Qubit: Two-Way Quantum Finite Automata and the Word Problem}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{139:1--139:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.139},
  URN =		{urn:nbn:de:0030-drops-125468},
  doi =		{10.4230/LIPIcs.ICALP.2020.139},
  annote =	{Keywords: finite automata, quantum, word problem of a group}
}
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  • 6 Theory of computation → Quantum complexity theory
  • 3 Theory of computation → Complexity classes
  • 2 Theory of computation → Formal languages and automata theory
  • 2 Theory of computation → Quantum computation theory
  • 1 Theory of computation → Circuit complexity
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  • 2 Quantum computation
  • 2 quantum complexity
  • 1 Finite automata
  • 1 Graph connectivity
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