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**Published in:** LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

We study the advantages of quantum communication models over classical communication models that are equipped with a limited number of qubits of entanglement. In this direction, we give explicit partial functions on n bits for which reducing the entanglement increases the classical communication complexity exponentially. Our separations are as follows. For every k ≥ ~1:
Q‖^* versus R2^*: We show that quantum simultaneous protocols with Θ̃(k⁵log³n) qubits of entanglement can exponentially outperform two-way randomized protocols with O(k) qubits of entanglement. This resolves an open problem from [Dmitry Gavinsky, 2008] and improves the state-of-the-art separations between quantum simultaneous protocols with entanglement and two-way randomized protocols without entanglement [Gavinsky, 2019; Girish et al., 2022].
R‖^* versus Q‖^*: We show that classical simultaneous protocols with Θ̃(k log n) qubits of entanglement can exponentially outperform quantum simultaneous protocols with O(k) qubits of entanglement, resolving an open question from [Gavinsky et al., 2006; Gavinsky, 2019]. The best result prior to our work was a relational separation against protocols without entanglement [Gavinsky et al., 2006].
R‖^* versus R1^*: We show that classical simultaneous protocols with Θ̃(k log n) qubits of entanglement can exponentially outperform randomized one-way protocols with O(k) qubits of entanglement. Prior to our work, only a relational separation was known [Dmitry Gavinsky, 2008].

Srinivasan Arunachalam and Uma Girish. Trade-Offs Between Entanglement and Communication. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 25:1-25:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{arunachalam_et_al:LIPIcs.CCC.2023.25, author = {Arunachalam, Srinivasan and Girish, Uma}, title = {{Trade-Offs Between Entanglement and Communication}}, booktitle = {38th Computational Complexity Conference (CCC 2023)}, pages = {25:1--25:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-282-2}, ISSN = {1868-8969}, year = {2023}, volume = {264}, editor = {Ta-Shma, Amnon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.25}, URN = {urn:nbn:de:0030-drops-182957}, doi = {10.4230/LIPIcs.CCC.2023.25}, annote = {Keywords: quantum, communication complexity, exponential separation, boolean hidden matching, forrelation, xor lemma} }

Document

**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Randomized algorithms and protocols assume the availability of a perfect source of randomness. In real life, however, perfect randomness is rare and is almost never guaranteed. The gap between these two facts motivated much of the work on randomness and derandomization in theoretical computer science.
In this work, we define a new type of randomized algorithms (and protocols), that we call robustly-randomized algorithms (protocols). Such algorithms have access to two separate (read-once) random strings. The first string is trusted to be perfectly random, but its length is bounded by some parameter k = k(n) (where n is the length of the input). We think of k as relatively small, say sub-linear or poly-logarithmic in n. The second string is of unbounded length and is assumed to be random, but its randomness is not trusted.
The output of the algorithm is either an output in the set of possible outputs of the problem, or a special symbol, interpreted as do not know and denoted by ⊥. On every input for the algorithm, the output of the algorithm must satisfy the following two requirements:
1) If the second random string is perfectly random then the algorithm must output the correct answer with high probability.
2) If the second random string is an arbitrary string, even adversarially chosen after seeing the input, the algorithm must output with high probability either the correct answer or the special symbol ⊥.
We discuss relations of this new definition to several previously studied notions in randomness and derandomization. For example, when considering polynomial-time algorithms, if k is logarithmic we get the complexity class ZPP, while if k is unbounded we get the complexity class BPP, and for a general k, the algorithm can be viewed as an interactive proof with a probabilistic polynomial-time prover and a probabilistic polynomial-time verifier, where the prover is allowed an unlimited number of random bits and the verifier is limited to at most k random bits.
Every previously-studied class of randomized algorithms or protocols, and more generally, every previous use of randomness in theoretical computer science, can be revisited and redefined in light of our new definition, by replacing each random string with a pair of random strings, the first is trusted to be perfectly random but is relatively short and the second is of unlimited length but its randomness is not trusted. The main question that we ask is: In which settings and for which problems is the untrusted random string helpful?
Our main technical observation is that every problem in the class BPL (of problems solvable by bounded-error randomized logspace algorithms) can be solved by a robustly-randomized logspace algorithm with k = O(log n), that is with just a logarithmic number of trusted random bits. We also give query complexity separations that show cases where the untrusted random string is provenly helpful. Specifically, we show that there are promise problems that can be solved by robustly-randomized protocols with only one query and just a logarithmic number of trusted random bits, whereas any randomized protocol requires either a linear number of random bits or an exponential number of queries, and any zero-error randomized protocol requires a polynomial number of queries.

Uma Girish, Ran Raz, and Wei Zhan. Is Untrusted Randomness Helpful?. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 56:1-56:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{girish_et_al:LIPIcs.ITCS.2023.56, author = {Girish, Uma and Raz, Ran and Zhan, Wei}, title = {{Is Untrusted Randomness Helpful?}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {56:1--56:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.56}, URN = {urn:nbn:de:0030-drops-175593}, doi = {10.4230/LIPIcs.ITCS.2023.56}, annote = {Keywords: Untrusted, Randomness, Verifiable, ZPL, BPL, ZPP, BPP} }

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RANDOM

**Published in:** LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

We prove that for every 3-player (3-prover) game G with value less than one, whose query distribution has the support S = {(1,0,0), (0,1,0), (0,0,1)} of Hamming weight one vectors, the value of the n-fold parallel repetition G^{⊗n} decays polynomially fast to zero; that is, there is a constant c = c(G) > 0 such that the value of the game G^{⊗n} is at most n^{-c}.
Following the recent work of Girish, Holmgren, Mittal, Raz and Zhan (STOC 2022), our result is the missing piece that implies a similar bound for a much more general class of multiplayer games: For every 3-player game G over binary questions and arbitrary answer lengths, with value less than 1, there is a constant c = c(G) > 0 such that the value of the game G^{⊗n} is at most n^{-c}.
Our proof technique is new and requires many new ideas. For example, we make use of the Level-k inequalities from Boolean Fourier Analysis, which, to the best of our knowledge, have not been explored in this context prior to our work.

Uma Girish, Kunal Mittal, Ran Raz, and Wei Zhan. Polynomial Bounds on Parallel Repetition for All 3-Player Games with Binary Inputs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{girish_et_al:LIPIcs.APPROX/RANDOM.2022.6, author = {Girish, Uma and Mittal, Kunal and Raz, Ran and Zhan, Wei}, title = {{Polynomial Bounds on Parallel Repetition for All 3-Player Games with Binary Inputs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {6:1--6:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.6}, URN = {urn:nbn:de:0030-drops-171286}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.6}, annote = {Keywords: Parallel repetition, Multi-prover games, Fourier analysis} }

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

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.

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} }

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RANDOM

**Published in:** LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

The Forrelation problem, first introduced by Aaronson [Scott Aaronson, 2010] and Aaronson and Ambainis [Scott Aaronson and Andris Ambainis, 2015], is a well studied computational problem in the context of separating quantum and classical computational models. Variants of this problem were used to give tight separations between quantum and classical query complexity [Scott Aaronson and Andris Ambainis, 2015]; the first separation between poly-logarithmic quantum query complexity and bounded-depth circuits of super-polynomial size, a result that also implied an oracle separation of the classes BQP and PH [Ran Raz and Avishay Tal, 2019]; and improved separations between quantum and classical communication complexity [Uma Girish et al., 2021]. In all these separations, the lower bound for the classical model only holds when the advantage of the protocol (over a random guess) is more than ≈ 1/√N, that is, the success probability is larger than ≈ 1/2 + 1/√N. This is unavoidable as ≈ 1/√N is the correlation between two coordinates of an input that is sampled from the Forrelation distribution, and hence there are simple classical protocols that achieve advantage ≈ 1/√N, in all these models.
To achieve separations when the classical protocol has smaller advantage, we study in this work the xor of k independent copies of (a variant of) the Forrelation function (where k≪ N). We prove a very general result that shows that any family of Boolean functions that is closed under restrictions, whose Fourier mass at level 2k is bounded by α^k (that is, the sum of the absolute values of all Fourier coefficients at level 2k is bounded by α^k), cannot compute the xor of k independent copies of the Forrelation function with advantage better than O((α^k)/(N^{k/2})). This is a strengthening of a result of [Eshan Chattopadhyay et al., 2019], that gave a similar statement for k = 1, using the technique of [Ran Raz and Avishay Tal, 2019]. We give several applications of our result. In particular, we obtain the following separations:
Quantum versus Classical Communication Complexity. We give the first example of a partial Boolean function that can be computed by a simultaneous-message quantum protocol with communication complexity polylog(N) (where Alice and Bob also share polylog(N) EPR pairs), and such that, any classical randomized protocol of communication complexity at most õ(N^{1/4}), with any number of rounds, has quasipolynomially small advantage over a random guess. Previously, only separations where the classical protocol has polynomially small advantage were known between these models [Dmitry Gavinsky, 2016; Uma Girish et al., 2021].
Quantum Query Complexity versus Bounded Depth Circuits. We give the first example of a partial Boolean function that has a quantum query algorithm with query complexity polylog(N), and such that, any constant-depth circuit of quasipolynomial size has quasipolynomially small advantage over a random guess. Previously, only separations where the constant-depth circuit has polynomially small advantage were known [Ran Raz and Avishay Tal, 2019].

Uma Girish, Ran Raz, and Wei Zhan. Lower Bounds for XOR of Forrelations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 52:1-52:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{girish_et_al:LIPIcs.APPROX/RANDOM.2021.52, author = {Girish, Uma and Raz, Ran and Zhan, Wei}, title = {{Lower Bounds for XOR of Forrelations}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {52:1--52:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-207-5}, ISSN = {1868-8969}, year = {2021}, volume = {207}, editor = {Wootters, Mary and Sanit\`{a}, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.52}, URN = {urn:nbn:de:0030-drops-147453}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.52}, annote = {Keywords: Forrelation, Quasipolynomial, Separation, Quantum versus Classical, Xor} }

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RANDOM

**Published in:** LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

We give a new proof of the fact that the parallel repetition of the (3-player) GHZ game reduces the value of the game to zero polynomially quickly. That is, we show that the value of the n-fold GHZ game is at most n^{-Ω(1)}. This was first established by Holmgren and Raz [Holmgren and Raz, 2020]. We present a new proof of this theorem that we believe to be simpler and more direct. Unlike most previous works on parallel repetition, our proof makes no use of information theory, and relies on the use of Fourier analysis.
The GHZ game [Greenberger et al., 1989] has played a foundational role in the understanding of quantum information theory, due in part to the fact that quantum strategies can win the GHZ game with probability 1. It is possible that improved parallel repetition bounds may find applications in this setting.
Recently, Dinur, Harsha, Venkat, and Yuen [Dinur et al., 2017] highlighted the GHZ game as a simple three-player game, which is in some sense maximally far from the class of multi-player games whose behavior under parallel repetition is well understood. Dinur et al. conjectured that parallel repetition decreases the value of the GHZ game exponentially quickly, and speculated that progress on proving this would shed light on parallel repetition for general multi-player (multi-prover) games.

Uma Girish, Justin Holmgren, Kunal Mittal, Ran Raz, and Wei Zhan. Parallel Repetition for the GHZ Game: A Simpler Proof. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 62:1-62:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{girish_et_al:LIPIcs.APPROX/RANDOM.2021.62, author = {Girish, Uma and Holmgren, Justin and Mittal, Kunal and Raz, Ran and Zhan, Wei}, title = {{Parallel Repetition for the GHZ Game: A Simpler Proof}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {62:1--62:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-207-5}, ISSN = {1868-8969}, year = {2021}, volume = {207}, editor = {Wootters, Mary and Sanit\`{a}, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.62}, URN = {urn:nbn:de:0030-drops-147551}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.62}, annote = {Keywords: Parallel Repetition, GHZ, Polynomial, Multi-player} }

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**Published in:** LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

We prove that for every parity decision tree of depth d on n variables, the sum of absolute values of Fourier coefficients at level 𝓁 is at most d^{𝓁/2} ⋅ O(𝓁 ⋅ log(n))^𝓁. Our result is nearly tight for small values of 𝓁 and extends a previous Fourier bound for standard decision trees by Sherstov, Storozhenko, and Wu (STOC, 2021).
As an application of our Fourier bounds, using the results of Bansal and Sinha (STOC, 2021), we show that the k-fold Forrelation problem has (randomized) parity decision tree complexity Ω̃(n^{1-1/k}), while having quantum query complexity ⌈ k/2⌉.
Our proof follows a random-walk approach, analyzing the contribution of a random path in the decision tree to the level-𝓁 Fourier expression. To carry the argument, we apply a careful cleanup procedure to the parity decision tree, ensuring that the value of the random walk is bounded with high probability. We observe that step sizes for the level-𝓁 walks can be computed by the intermediate values of level ≤ 𝓁-1 walks, which calls for an inductive argument. Our approach differs from previous proofs of Tal (FOCS, 2020) and Sherstov, Storozhenko, and Wu (STOC, 2021) that relied on decompositions of the tree. In particular, for the special case of standard decision trees we view our proof as slightly simpler and more intuitive.
In addition, we prove a similar bound for noisy decision trees of cost at most d - a model that was recently introduced by Ben-David and Blais (FOCS, 2020).

Uma Girish, Avishay Tal, and Kewen Wu. Fourier Growth of Parity Decision Trees. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 39:1-39:36, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{girish_et_al:LIPIcs.CCC.2021.39, author = {Girish, Uma and Tal, Avishay and Wu, Kewen}, title = {{Fourier Growth of Parity Decision Trees}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {39:1--39:36}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-193-1}, ISSN = {1868-8969}, year = {2021}, volume = {200}, editor = {Kabanets, Valentine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.39}, URN = {urn:nbn:de:0030-drops-143137}, doi = {10.4230/LIPIcs.CCC.2021.39}, annote = {Keywords: Fourier analysis of Boolean functions, noisy decision tree, parity decision tree, query complexity} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

We give a quantum logspace algorithm for powering contraction matrices, that is, matrices with spectral norm at most 1. The algorithm gets as an input an arbitrary n× n contraction matrix A, and a parameter T ≤ poly(n) and outputs the entries of A^T, up to (arbitrary) polynomially small additive error. The algorithm applies only unitary operators, without intermediate measurements. We show various implications and applications of this result:
First, we use this algorithm to show that the class of quantum logspace algorithms with only quantum memory and with intermediate measurements is equivalent to the class of quantum logspace algorithms with only quantum memory without intermediate measurements. This shows that the deferred-measurement principle, a fundamental principle of quantum computing, applies also for quantum logspace algorithms (without classical memory). More generally, we give a quantum algorithm with space O(S + log T) that takes as an input the description of a quantum algorithm with quantum space S and time T, with intermediate measurements (without classical memory), and simulates it unitarily with polynomially small error, without intermediate measurements.
Since unitary transformations are reversible (while measurements are irreversible) an interesting aspect of this result is that it shows that any quantum logspace algorithm (without classical memory) can be simulated by a reversible quantum logspace algorithm. This proves a quantum analogue of the result of Lange, McKenzie and Tapp that deterministic logspace is equal to reversible logspace [Lange et al., 2000].
Finally, we use our results to show non-trivial classical simulations of quantum logspace learning algorithms.

Uma Girish, Ran Raz, and Wei Zhan. Quantum Logspace Algorithm for Powering Matrices with Bounded Norm. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 73:1-73:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{girish_et_al:LIPIcs.ICALP.2021.73, author = {Girish, Uma and Raz, Ran and Zhan, Wei}, title = {{Quantum Logspace Algorithm for Powering Matrices with Bounded Norm}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {73:1--73:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.73}, URN = {urn:nbn:de:0030-drops-141426}, doi = {10.4230/LIPIcs.ICALP.2021.73}, annote = {Keywords: BQL, Matrix Powering, Quantum Circuit, Reversible Computation} }

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

We study a new type of separations between quantum and classical communication complexity, separations that are obtained using quantum protocols where all parties are efficient, in the sense that they can be implemented by small quantum circuits, with oracle access to their inputs. Our main result qualitatively matches the strongest known separation between quantum and classical communication complexity [Dmitry Gavinsky, 2016] and is obtained using a quantum protocol where all parties are efficient. More precisely, we give an explicit partial Boolean function f over inputs of length N, such that:
(1) f can be computed by a simultaneous-message quantum protocol with communication complexity polylog(N) (where at the beginning of the protocol Alice and Bob also have polylog(N) entangled EPR pairs).
(2) Any classical randomized protocol for f, with any number of rounds, has communication complexity at least Ω̃(N^{1/4}).
(3) All parties in the quantum protocol of Item (1) (Alice, Bob and the referee) can be implemented by quantum circuits of size polylog(N) (where Alice and Bob have oracle access to their inputs).
Items (1), (2) qualitatively match the strongest known separation between quantum and classical communication complexity, proved by Gavinsky [Dmitry Gavinsky, 2016]. Item (3) is new. (Our result is incomparable to the one of Gavinsky. While he obtained a quantitatively better lower bound of Ω(N^{1/2}) in the classical case, the referee in his quantum protocol is inefficient).
Exponential separations of quantum and classical communication complexity have been studied in numerous previous works, but to the best of our knowledge the efficiency of the parties in the quantum protocol has not been addressed, and in most previous separations the quantum parties seem to be inefficient. The only separations that we know of that have efficient quantum parties are the recent separations that are based on lifting [Arkadev Chattopadhyay et al., 2019; Arkadev Chattopadhyay et al., 2019]. However, these separations seem to require quantum protocols with at least two rounds of communication, so they imply a separation of two-way quantum and classical communication complexity but they do not give the stronger separations of simultaneous-message quantum communication complexity vs. two-way classical communication complexity (or even one-way quantum communication complexity vs. two-way classical communication complexity).
Our proof technique is completely new, in the context of communication complexity, and is based on techniques from [Ran Raz and Avishay Tal, 2019]. Our function f is based on a lift of the forrelation problem, using xor as a gadget.

Uma Girish, Ran Raz, and Avishay Tal. Quantum Versus Randomized Communication Complexity, with Efficient Players. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 54:1-54:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{girish_et_al:LIPIcs.ITCS.2021.54, author = {Girish, Uma and Raz, Ran and Tal, Avishay}, title = {{Quantum Versus Randomized Communication Complexity, with Efficient Players}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {54:1--54:20}, 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.2021.54}, URN = {urn:nbn:de:0030-drops-135932}, doi = {10.4230/LIPIcs.ITCS.2021.54}, annote = {Keywords: Exponential Separation, Quantum, Randomized, Communication, Complexity, Forrelation} }