Ansis Rosmanis - Academia.edu (original) (raw)

Papers by Ansis Rosmanis

arXiv (Cornell University), Oct 18, 2013

We prove tight Ω(n 1/3) lower bounds on the quantum query complexity of the Collision and the Set... more We prove tight Ω(n 1/3) lower bounds on the quantum query complexity of the Collision and the Set Equality problems, provided that the size of the alphabet is large enough. We do this using the negative-weight adversary method. Thus, we reprove the result by Aaronson and Shi, as well as a more recent development by Zhandry.

Linear Algebra and its Applications, Oct 1, 2012

We examine the fixed space of positive trace-preserving super-operators. We describe a specific s... more We examine the fixed space of positive trace-preserving super-operators. We describe a specific structure that this space must have and what the projection onto it must look like. We show how these results, in turn, lead to an alternative proof of the complete characterization of the fixed space of completely positive trace-preserving super-operators.

Conference on Innovations in Theoretical Computer Science, 2020

The index erasure problem is a quantum state generation problem that asks a quantum computer to p... more The index erasure problem is a quantum state generation problem that asks a quantum computer to prepare a uniform superposition over the image of an injective function given by an oracle. We prove a tight Ω(√ n) lower bound on the quantum query complexity of the noncoherent case of the problem, where, in addition to preparing the required superposition, the algorithm is allowed to leave the ancillary memory in an arbitrary function-dependent state. This resolves an open question of Ambainis et al., who gave a tight bound for the coherent case, the case where the ancillary memory must return to its initial state. To prove our main result, we first extend the automorphism principle of Høyer et al. to the general adversary method of Lee et al. for state generation problems, which allows one to exploit the symmetries of these problems to lower bound their quantum query complexity. Using this method, we establish a strong connection between the quantum query complexity of non-coherent symmetric state generation problems and the Krein parameters of an association scheme defined on injective functions. In particular, we use the spherical harmonics a finite symmetric Gelfand pair associated with the space of injective functions to obtain asymptotic bounds on certain Krein parameters, from which the main result follows.

Conference on Theory of Quantum Computation, Communication and Cryptography, Jun 1, 2020

Classical coupon collector: learn k-set S ⊆ [n] from Θ(k log k) uniform samples Quantum coupon co... more Classical coupon collector: learn k-set S ⊆ [n] from Θ(k log k) uniform samples Quantum coupon collector: learn k-set S ⊆ [n] from Θ(k log(m + 1)) uniform superpositions (m = n − k is number of missing items) We also gave tight bounds for learning S from copies of |S and reflections through |S Open problem: are the quantum sample complexities of proper and improper learning the same for all C?

arXiv (Cornell University), Apr 13, 2017

We introduce a definition of the fidelity function for multi-round quantum strategies, which we c... more We introduce a definition of the fidelity function for multi-round quantum strategies, which we call the strategy fidelity, that is a generalization of the fidelity function for quantum states. We provide many properties of the strategy fidelity including a Fuchs-van de Graaf relationship with the strategy norm. We also provide a general monotonicity result for both the strategy fidelity and strategy norm under the actions of strategy-to-strategy linear maps. We illustrate an operational interpretation of the strategy fidelity in the spirit of Uhlmann's Theorem and discuss its application to the security analysis of quantum protocols for interactive cryptographic tasks such as bit-commitment and oblivious string transfer. Our analysis is general in the sense that the actions of the protocol need not be fully specified, which is in stark contrast to most other security proofs. Lastly, we provide a semidefinite programming formulation of the strategy fidelity.

Quantum zero-knowledge proofs and quantum proofs of knowledge are inherently difficult to analyze... more Quantum zero-knowledge proofs and quantum proofs of knowledge are inherently difficult to analyze because their security analysis uses rewinding. Certain cases of quantum rewinding are handled by the results by Watrous (SIAM J Comput, 2009) and Unruh (Eurocrypt 2012), yet in general the problem remains elusive. We show that this is not only due to a lack of proof techniques: relative to an oracle, we show that classically secure proofs and proofs of knowledge are insecure in the quantum setting. More specifically, sigma-protocols, the Fiat-Shamir construction, and Fischlin's proof system are quantum insecure under assumptions that are sufficient for classical security. Additionally, we show that for similar reasons, computationally binding commitments provide almost no security guarantees in a quantum setting. To show these results, we develop the "pick-one trick", a general technique that allows an adversary to find one value satisfying a given predicate, but not two.

Quantum Information & Computation, May 1, 2022

The index erasure problem is a quantum state generation problem that asks a quantum computer to p... more The index erasure problem is a quantum state generation problem that asks a quantum computer to prepare a uniform superposition over the image of an injective function given by an oracle. We prove a tight Ω(√ n) lower bound on the quantum query complexity of the non-coherent case of the problem, where, in addition to preparing the required superposition, the algorithm is allowed to leave the ancillary memory in an arbitrary function-dependent state. This resolves an open question of Ambainis, Magnin, Roetteler, and Roland (CCC 2011), who gave a tight bound for the coherent case, the case where the ancillary memory must return to its initial state. To prove our main result, we first extend the so-called automorphism principle (Høyer et al. STOC 2007) to the general adversary method for state conversion problems (Lee et al. STOC 2011), which allows one to exploit the symmetries of these problems to lower bound their quantum query complexity. Using this method, we establish a strong connection between the quantum query complexity of non-coherent symmetric state generation problems and the well-known Krein parameters of association schemes. Krein parameters are usually hard to determine, nevertheless, we give a novel way of computing certain Krein parameters of a commutative association scheme defined over partial permutations. We believe the study of this association scheme may also be of independent interest.

arXiv (Cornell University), Jun 17, 2011

We examine the fixed space of positive trace preserving and completely positive trace preserving ... more We examine the fixed space of positive trace preserving and completely positive trace preserving super-operators. We describe what form the fixed space of a completely positive trace preserving super-operator must take, what dimensions this space may have, and what the projection onto it must look like.

arXiv (Cornell University), Mar 16, 2021

In his seminal work on recording quantum queries [Crypto 2019], Zhandry studied interactions betw... more In his seminal work on recording quantum queries [Crypto 2019], Zhandry studied interactions between quantum query algorithms and the quantum oracle corresponding to random functions. Zhandry presented a framework for interpreting various states in the quantum space of the oracle as databases of the knowledge acquired by the algorithm and used that interpretation to provide security proofs in post-quantum cryptography. In this paper, we introduce a similar interpretation for the case when the oracle corresponds to random permutations instead of random functions. Because both random functions and random permutations are highly significant in security proofs, we hope that the present framework will find applications in quantum cryptography. Additionally, we show how this framework can be used to prove that the success probability for a κ-query quantum algorithm that attempts to invert a random Nelement permutation is at most O(κ 2 /N).

Quantum Information & Computation, Mar 1, 2018

We prove tight Ω(n 1/3) lower bounds on the quantum query complexity of the Collision and the Set... more We prove tight Ω(n 1/3) lower bounds on the quantum query complexity of the Collision and the Set Equality problems, provided that the size of the alphabet is large enough. We do this using the negative-weight adversary method. Thus, we reprove the result by Aaronson and Shi, as well as a more recent development by Zhandry.

arXiv (Cornell University), Jan 15, 2014

Physical Review A, Feb 7, 2011

I introduce a new type of continuous-time quantum walk on graphs called the quantum snake walk, t... more I introduce a new type of continuous-time quantum walk on graphs called the quantum snake walk, the basis states of which are fixed-length paths (snakes) in the underlying graph. First I analyze the quantum snake walk on the line, and I show that, even though most states stay localized throughout the evolution, there are specific states which most likely move on the line as wave packets with momentum inversely proportional to the length of the snake. Next I discuss how an algorithm based on the quantum snake walk might potentially be able to solve an extended version of the glued trees problem which asks to find a path connecting both roots of the glued trees graph. No efficient quantum algorithm solving this problem is known yet.

Quantum Information & Computation, Dec 1, 2019

Submodular functions are set functions mapping every subset of some ground set of size n into the... more Submodular functions are set functions mapping every subset of some ground set of size n into the real numbers and satisfying the diminishing returns property. Submodular minimization is an important field in discrete optimization theory due to its relevance for various branches of mathematics, computer science and economics. The currently fastest strongly polynomial algorithm for exact minimization [LSW15] runs in time O(n 3 • EO + n 4) where EO denotes the cost to evaluate the function on any set. For functions with range [−1, 1], the best ǫ-additive approximation algorithm [CLSW17] runs in time O(n 5/3 /ǫ 2 • EO). In this paper we present a classical and a quantum algorithm for approximate submodular minimization. Our classical result improves on the algorithm of [CLSW17] and runs in time O(n 3/2 /ǫ 2 • EO). Our quantum algorithm is, up to our knowledge, the first attempt to use quantum computing for submodular optimization. The algorithm runs in time O(n 5/4 /ǫ 5/2 • log(1/ǫ) • EO). The main ingredient of the quantum result is a new method for sampling with high probability T independent elements from any discrete probability distribution of support size n in time O(√ T n). Previous quantum algorithms for this problem were of complexity O(T √ n).

Energy and Buildings, Mar 1, 2022

Lighting load accounts for a significant portion of overall energy consumption in office building... more Lighting load accounts for a significant portion of overall energy consumption in office buildings. To reduce this load, we have designed and built a smart self-calibrating lighting control system that minimizes power consumption that automatically responds to changes in daylight and occupancy, while simultaneously providing personalized lighting comfort to each occupant. The system measures illuminance and occupancy from sensors located at each work station. Using an unobtrusive self-calibration process, it estimates the relationship between the dimming level of each bulb and the illuminance at each work station. Subsequently, an adaptive control algorithm maintains the desired illuminance at work surfaces despite environmental fluctuations by periodically recalculating the power-efficient and comfort-preserving dimming level for each bulb. Based on a realistic deployment of our system, we find that our system quickly responds to changes in occupancy, daylight and user preferences. We also show, through extensive simulations using 7 months of collected daylight and occupancy data, that our system reduces energy consumption by about 40% compared to conventional LED lighting systems.

arXiv (Cornell University), Feb 18, 2020

arXiv (Cornell University), Dec 29, 2017

In this paper, we study quantum query complexity of the following rather natural tripartite gener... more In this paper, we study quantum query complexity of the following rather natural tripartite generalisations (in the spirit of the 3-sum problem) of the hidden shift and the set equality problems, which we call the 3-shift-sum and the 3-matching-sum problems. The 3-shift-sum problem is as follows: given a table of 3 × n elements, is it possible to circularly shift its rows so that the sum of the elements in each column becomes zero? It is promised that, if this is not the case, then no 3 elements in the table sum up to zero. The 3-matching-sum problem is defined similarly, but it is allowed to arbitrarily permute elements within each row. For these problems, we prove lower bounds of Ω(n 1/3) and Ω(√ n), respectively. The second lower bound is tight. The lower bounds are proven by a novel application of the dual learning graph framework and by using representation-theoretic tools from [6].

arXiv (Cornell University), Feb 19, 2019

The index erasure problem is a quantum state generation problem that asks a quantum computer to p... more The index erasure problem is a quantum state generation problem that asks a quantum computer to prepare a uniform superposition over the image of an injective function given by an oracle. We prove a tight Ω(√ n) lower bound on the quantum query complexity of the noncoherent case of the problem, where, in addition to preparing the required superposition, the algorithm is allowed to leave the ancillary memory in an arbitrary function-dependent state. This resolves an open question of Ambainis et al., who gave a tight bound for the coherent case, the case where the ancillary memory must return to its initial state. To prove our main result, we first extend the automorphism principle of Høyer et al. to the general adversary method of Lee et al. for state generation problems, which allows one to exploit the symmetries of these problems to lower bound their quantum query complexity. Using this method, we establish a strong connection between the quantum query complexity of non-coherent symmetric state generation problems and the Krein parameters of an association scheme defined on injective functions. In particular, we use the spherical harmonics a finite symmetric Gelfand pair associated with the space of injective functions to obtain asymptotic bounds on certain Krein parameters, from which the main result follows.

Chicago Journal of Theoretical Computer Science, 2014

The ELEMENT DISTINCTNESS problem is to decide whether each character of an input string is unique... more The ELEMENT DISTINCTNESS problem is to decide whether each character of an input string is unique. The quantum query complexity of ELEMENT DISTINCTNESS is known to be Θ(N 2/3); the polynomial method gives a tight lower bound for any input alphabet, while a tight adversary construction was only known for alphabets of size Ω(N 2). We construct a tight Ω(N 2/3) adversary lower bound for ELEMENT DISTINCTNESS with minimal non-trivial alphabet size, which equals the length of the input. This result may help to improve lower bounds for other related query problems.

Symposium on Theoretical Aspects of Computer Science, 2019

There are two central models considered in (fault-free synchronous) distributed computing: the CO... more There are two central models considered in (fault-free synchronous) distributed computing: the CONGEST model, in which communication channels have limited bandwidth, and the LOCAL model, in which communication channels have unlimited bandwidth. Very recently, Le Gall and Magniez (PODC 2018) showed the superiority of quantum distributed computing over classical distributed computing in the CONGEST model. In this work we show the superiority of quantum distributed computing in the LOCAL model: we exhibit a computational task that can be solved in a constant number of rounds in the quantum setting but requires Ωpnq rounds in the classical setting, where n denotes the size of the network.

arXiv (Cornell University), Jun 16, 2011

We examine the fixed space of positive trace-preserving super-operators. We describe a specific s... more We examine the fixed space of positive trace-preserving super-operators. We describe a specific structure that this space must have and what the projection onto it must look like. We show how these results, in turn, lead to an alternative proof of the complete characterization of the fixed space of completely positive trace-preserving super-operators.

arXiv (Cornell University), Oct 18, 2013

We prove tight Ω(n 1/3) lower bounds on the quantum query complexity of the Collision and the Set... more We prove tight Ω(n 1/3) lower bounds on the quantum query complexity of the Collision and the Set Equality problems, provided that the size of the alphabet is large enough. We do this using the negative-weight adversary method. Thus, we reprove the result by Aaronson and Shi, as well as a more recent development by Zhandry.

Linear Algebra and its Applications, Oct 1, 2012

We examine the fixed space of positive trace-preserving super-operators. We describe a specific s... more We examine the fixed space of positive trace-preserving super-operators. We describe a specific structure that this space must have and what the projection onto it must look like. We show how these results, in turn, lead to an alternative proof of the complete characterization of the fixed space of completely positive trace-preserving super-operators.

Conference on Innovations in Theoretical Computer Science, 2020

The index erasure problem is a quantum state generation problem that asks a quantum computer to p... more The index erasure problem is a quantum state generation problem that asks a quantum computer to prepare a uniform superposition over the image of an injective function given by an oracle. We prove a tight Ω(√ n) lower bound on the quantum query complexity of the noncoherent case of the problem, where, in addition to preparing the required superposition, the algorithm is allowed to leave the ancillary memory in an arbitrary function-dependent state. This resolves an open question of Ambainis et al., who gave a tight bound for the coherent case, the case where the ancillary memory must return to its initial state. To prove our main result, we first extend the automorphism principle of Høyer et al. to the general adversary method of Lee et al. for state generation problems, which allows one to exploit the symmetries of these problems to lower bound their quantum query complexity. Using this method, we establish a strong connection between the quantum query complexity of non-coherent symmetric state generation problems and the Krein parameters of an association scheme defined on injective functions. In particular, we use the spherical harmonics a finite symmetric Gelfand pair associated with the space of injective functions to obtain asymptotic bounds on certain Krein parameters, from which the main result follows.

Conference on Theory of Quantum Computation, Communication and Cryptography, Jun 1, 2020

Classical coupon collector: learn k-set S ⊆ [n] from Θ(k log k) uniform samples Quantum coupon co... more Classical coupon collector: learn k-set S ⊆ [n] from Θ(k log k) uniform samples Quantum coupon collector: learn k-set S ⊆ [n] from Θ(k log(m + 1)) uniform superpositions (m = n − k is number of missing items) We also gave tight bounds for learning S from copies of |S and reflections through |S Open problem: are the quantum sample complexities of proper and improper learning the same for all C?

arXiv (Cornell University), Apr 13, 2017

We introduce a definition of the fidelity function for multi-round quantum strategies, which we c... more We introduce a definition of the fidelity function for multi-round quantum strategies, which we call the strategy fidelity, that is a generalization of the fidelity function for quantum states. We provide many properties of the strategy fidelity including a Fuchs-van de Graaf relationship with the strategy norm. We also provide a general monotonicity result for both the strategy fidelity and strategy norm under the actions of strategy-to-strategy linear maps. We illustrate an operational interpretation of the strategy fidelity in the spirit of Uhlmann's Theorem and discuss its application to the security analysis of quantum protocols for interactive cryptographic tasks such as bit-commitment and oblivious string transfer. Our analysis is general in the sense that the actions of the protocol need not be fully specified, which is in stark contrast to most other security proofs. Lastly, we provide a semidefinite programming formulation of the strategy fidelity.

Quantum zero-knowledge proofs and quantum proofs of knowledge are inherently difficult to analyze... more Quantum zero-knowledge proofs and quantum proofs of knowledge are inherently difficult to analyze because their security analysis uses rewinding. Certain cases of quantum rewinding are handled by the results by Watrous (SIAM J Comput, 2009) and Unruh (Eurocrypt 2012), yet in general the problem remains elusive. We show that this is not only due to a lack of proof techniques: relative to an oracle, we show that classically secure proofs and proofs of knowledge are insecure in the quantum setting. More specifically, sigma-protocols, the Fiat-Shamir construction, and Fischlin's proof system are quantum insecure under assumptions that are sufficient for classical security. Additionally, we show that for similar reasons, computationally binding commitments provide almost no security guarantees in a quantum setting. To show these results, we develop the "pick-one trick", a general technique that allows an adversary to find one value satisfying a given predicate, but not two.

Quantum Information & Computation, May 1, 2022

The index erasure problem is a quantum state generation problem that asks a quantum computer to p... more The index erasure problem is a quantum state generation problem that asks a quantum computer to prepare a uniform superposition over the image of an injective function given by an oracle. We prove a tight Ω(√ n) lower bound on the quantum query complexity of the non-coherent case of the problem, where, in addition to preparing the required superposition, the algorithm is allowed to leave the ancillary memory in an arbitrary function-dependent state. This resolves an open question of Ambainis, Magnin, Roetteler, and Roland (CCC 2011), who gave a tight bound for the coherent case, the case where the ancillary memory must return to its initial state. To prove our main result, we first extend the so-called automorphism principle (Høyer et al. STOC 2007) to the general adversary method for state conversion problems (Lee et al. STOC 2011), which allows one to exploit the symmetries of these problems to lower bound their quantum query complexity. Using this method, we establish a strong connection between the quantum query complexity of non-coherent symmetric state generation problems and the well-known Krein parameters of association schemes. Krein parameters are usually hard to determine, nevertheless, we give a novel way of computing certain Krein parameters of a commutative association scheme defined over partial permutations. We believe the study of this association scheme may also be of independent interest.

arXiv (Cornell University), Jun 17, 2011

We examine the fixed space of positive trace preserving and completely positive trace preserving ... more We examine the fixed space of positive trace preserving and completely positive trace preserving super-operators. We describe what form the fixed space of a completely positive trace preserving super-operator must take, what dimensions this space may have, and what the projection onto it must look like.

arXiv (Cornell University), Mar 16, 2021

In his seminal work on recording quantum queries [Crypto 2019], Zhandry studied interactions betw... more In his seminal work on recording quantum queries [Crypto 2019], Zhandry studied interactions between quantum query algorithms and the quantum oracle corresponding to random functions. Zhandry presented a framework for interpreting various states in the quantum space of the oracle as databases of the knowledge acquired by the algorithm and used that interpretation to provide security proofs in post-quantum cryptography. In this paper, we introduce a similar interpretation for the case when the oracle corresponds to random permutations instead of random functions. Because both random functions and random permutations are highly significant in security proofs, we hope that the present framework will find applications in quantum cryptography. Additionally, we show how this framework can be used to prove that the success probability for a κ-query quantum algorithm that attempts to invert a random Nelement permutation is at most O(κ 2 /N).

Quantum Information & Computation, Mar 1, 2018

We prove tight Ω(n 1/3) lower bounds on the quantum query complexity of the Collision and the Set... more We prove tight Ω(n 1/3) lower bounds on the quantum query complexity of the Collision and the Set Equality problems, provided that the size of the alphabet is large enough. We do this using the negative-weight adversary method. Thus, we reprove the result by Aaronson and Shi, as well as a more recent development by Zhandry.

arXiv (Cornell University), Jan 15, 2014

Physical Review A, Feb 7, 2011

I introduce a new type of continuous-time quantum walk on graphs called the quantum snake walk, t... more I introduce a new type of continuous-time quantum walk on graphs called the quantum snake walk, the basis states of which are fixed-length paths (snakes) in the underlying graph. First I analyze the quantum snake walk on the line, and I show that, even though most states stay localized throughout the evolution, there are specific states which most likely move on the line as wave packets with momentum inversely proportional to the length of the snake. Next I discuss how an algorithm based on the quantum snake walk might potentially be able to solve an extended version of the glued trees problem which asks to find a path connecting both roots of the glued trees graph. No efficient quantum algorithm solving this problem is known yet.

Quantum Information & Computation, Dec 1, 2019

Submodular functions are set functions mapping every subset of some ground set of size n into the... more Submodular functions are set functions mapping every subset of some ground set of size n into the real numbers and satisfying the diminishing returns property. Submodular minimization is an important field in discrete optimization theory due to its relevance for various branches of mathematics, computer science and economics. The currently fastest strongly polynomial algorithm for exact minimization [LSW15] runs in time O(n 3 • EO + n 4) where EO denotes the cost to evaluate the function on any set. For functions with range [−1, 1], the best ǫ-additive approximation algorithm [CLSW17] runs in time O(n 5/3 /ǫ 2 • EO). In this paper we present a classical and a quantum algorithm for approximate submodular minimization. Our classical result improves on the algorithm of [CLSW17] and runs in time O(n 3/2 /ǫ 2 • EO). Our quantum algorithm is, up to our knowledge, the first attempt to use quantum computing for submodular optimization. The algorithm runs in time O(n 5/4 /ǫ 5/2 • log(1/ǫ) • EO). The main ingredient of the quantum result is a new method for sampling with high probability T independent elements from any discrete probability distribution of support size n in time O(√ T n). Previous quantum algorithms for this problem were of complexity O(T √ n).

Energy and Buildings, Mar 1, 2022

Lighting load accounts for a significant portion of overall energy consumption in office building... more Lighting load accounts for a significant portion of overall energy consumption in office buildings. To reduce this load, we have designed and built a smart self-calibrating lighting control system that minimizes power consumption that automatically responds to changes in daylight and occupancy, while simultaneously providing personalized lighting comfort to each occupant. The system measures illuminance and occupancy from sensors located at each work station. Using an unobtrusive self-calibration process, it estimates the relationship between the dimming level of each bulb and the illuminance at each work station. Subsequently, an adaptive control algorithm maintains the desired illuminance at work surfaces despite environmental fluctuations by periodically recalculating the power-efficient and comfort-preserving dimming level for each bulb. Based on a realistic deployment of our system, we find that our system quickly responds to changes in occupancy, daylight and user preferences. We also show, through extensive simulations using 7 months of collected daylight and occupancy data, that our system reduces energy consumption by about 40% compared to conventional LED lighting systems.

arXiv (Cornell University), Feb 18, 2020

arXiv (Cornell University), Dec 29, 2017

In this paper, we study quantum query complexity of the following rather natural tripartite gener... more In this paper, we study quantum query complexity of the following rather natural tripartite generalisations (in the spirit of the 3-sum problem) of the hidden shift and the set equality problems, which we call the 3-shift-sum and the 3-matching-sum problems. The 3-shift-sum problem is as follows: given a table of 3 × n elements, is it possible to circularly shift its rows so that the sum of the elements in each column becomes zero? It is promised that, if this is not the case, then no 3 elements in the table sum up to zero. The 3-matching-sum problem is defined similarly, but it is allowed to arbitrarily permute elements within each row. For these problems, we prove lower bounds of Ω(n 1/3) and Ω(√ n), respectively. The second lower bound is tight. The lower bounds are proven by a novel application of the dual learning graph framework and by using representation-theoretic tools from [6].

arXiv (Cornell University), Feb 19, 2019

The index erasure problem is a quantum state generation problem that asks a quantum computer to p... more The index erasure problem is a quantum state generation problem that asks a quantum computer to prepare a uniform superposition over the image of an injective function given by an oracle. We prove a tight Ω(√ n) lower bound on the quantum query complexity of the noncoherent case of the problem, where, in addition to preparing the required superposition, the algorithm is allowed to leave the ancillary memory in an arbitrary function-dependent state. This resolves an open question of Ambainis et al., who gave a tight bound for the coherent case, the case where the ancillary memory must return to its initial state. To prove our main result, we first extend the automorphism principle of Høyer et al. to the general adversary method of Lee et al. for state generation problems, which allows one to exploit the symmetries of these problems to lower bound their quantum query complexity. Using this method, we establish a strong connection between the quantum query complexity of non-coherent symmetric state generation problems and the Krein parameters of an association scheme defined on injective functions. In particular, we use the spherical harmonics a finite symmetric Gelfand pair associated with the space of injective functions to obtain asymptotic bounds on certain Krein parameters, from which the main result follows.

Chicago Journal of Theoretical Computer Science, 2014

The ELEMENT DISTINCTNESS problem is to decide whether each character of an input string is unique... more The ELEMENT DISTINCTNESS problem is to decide whether each character of an input string is unique. The quantum query complexity of ELEMENT DISTINCTNESS is known to be Θ(N 2/3); the polynomial method gives a tight lower bound for any input alphabet, while a tight adversary construction was only known for alphabets of size Ω(N 2). We construct a tight Ω(N 2/3) adversary lower bound for ELEMENT DISTINCTNESS with minimal non-trivial alphabet size, which equals the length of the input. This result may help to improve lower bounds for other related query problems.

Symposium on Theoretical Aspects of Computer Science, 2019

There are two central models considered in (fault-free synchronous) distributed computing: the CO... more There are two central models considered in (fault-free synchronous) distributed computing: the CONGEST model, in which communication channels have limited bandwidth, and the LOCAL model, in which communication channels have unlimited bandwidth. Very recently, Le Gall and Magniez (PODC 2018) showed the superiority of quantum distributed computing over classical distributed computing in the CONGEST model. In this work we show the superiority of quantum distributed computing in the LOCAL model: we exhibit a computational task that can be solved in a constant number of rounds in the quantum setting but requires Ωpnq rounds in the classical setting, where n denotes the size of the network.

arXiv (Cornell University), Jun 16, 2011

We examine the fixed space of positive trace-preserving super-operators. We describe a specific s... more We examine the fixed space of positive trace-preserving super-operators. We describe a specific structure that this space must have and what the projection onto it must look like. We show how these results, in turn, lead to an alternative proof of the complete characterization of the fixed space of completely positive trace-preserving super-operators.