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Quantum CopyProtection and Quantum Money
"... Forty years ago, Wiesner proposed using quantum states to create money that is physically impossible to counterfeit, something that cannot be done in the classical world. However, Wiesner’s scheme required a central bank to verify the money, and the question of whether there can be unclonable quantu ..."
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Forty years ago, Wiesner proposed using quantum states to create money that is physically impossible to counterfeit, something that cannot be done in the classical world. However, Wiesner’s scheme required a central bank to verify the money, and the question of whether there can be unclonable quantum money that anyone can verify has remained open since. One can also ask a related question, which seems to be new: can quantum states be used as copyprotected programs, which let the user evaluate some function f, but not create more programs for f? This paper tackles both questions using the arsenal of modern computational complexity. Our main result is that there exist quantum oracles relative to which publiclyverifiable quantum money is possible, and any family of functions that cannot be efficiently learned from its inputoutput behavior can be quantumly copyprotected. This provides the first formal evidence that these tasks are achievable. The technical core of our result is a “ComplexityTheoretic NoCloning Theorem,” which generalizes both the standard NoCloning Theorem and the optimality of Grover search, and might be of independent interest. Our security argument also requires explicit constructions of quantum tdesigns. Moving beyond the oracle world, we also present an explicit candidate scheme for publiclyverifiable quantum money, based on random stabilizer states; as well as two explicit schemes for copyprotecting the family of point functions. We do not know how to base the security of these schemes on any existing cryptographic assumption. (Note that without an oracle, we can only hope for security under some computational assumption.)
Code Generator for Quantum Simulated Annealing
"... This paper introduces QuSAnn v1.2 and Multiplexor Expander v1.2, two Java applications available for free. (Source code included in the distribution.) QuSAnn is a “code generator ” for quantum simulated annealing: after the user inputs some parameters, it outputs a quantum circuit for performing si ..."
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This paper introduces QuSAnn v1.2 and Multiplexor Expander v1.2, two Java applications available for free. (Source code included in the distribution.) QuSAnn is a “code generator ” for quantum simulated annealing: after the user inputs some parameters, it outputs a quantum circuit for performing simulated annealing on a quantum computer. The quantum circuit implements the algorithm of Wocjan et al. (arXiv:0804.4259), which improves on the original algorithm of Somma et al. (arXiv:0712.1008). The quantum circuit generated by QuSAnn includes some quantum multiplexors. The application Multiplexor Expander allows the user to replace each of those multiplexors by a sequence of more elementary gates such as multiply controlled NOTs and qubit rotations. 1
Bounds for error reduction with few quantum queries
 9th International Workshop on Randomization and Computation
, 2005
"... Abstract. We consider the quantum database search problem, where we are given a function f: [N] → {0, 1}, and are required to return an x ∈ [N] (a target address) such that f(x) = 1. Recently, Grover [G05] showed that there is an algorithm that after making one quantum query to the database, retur ..."
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Cited by 7 (2 self)
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Abstract. We consider the quantum database search problem, where we are given a function f: [N] → {0, 1}, and are required to return an x ∈ [N] (a target address) such that f(x) = 1. Recently, Grover [G05] showed that there is an algorithm that after making one quantum query to the database, returns an X ∈ [N] (a random variable) such that Pr[f(X) = 0] = ɛ 3, where ɛ = f −1 (0)/N. Using the same idea, Grover derived a tquery quantum algorithm (for infinitely many t) that errs with probability only ɛ 2t+1. Subsequently, Tulsi, Grover and Patel [TGP05] showed, using a different algorithm, that such a reduction can be achieved for all t. This method can be placed in a more general framework, where given any algorithm that produces a target state for some database f with probability of error ɛ, one can obtain another that makes t queries to f, and errs with probability ɛ 2t+1. For this method to work, we do not require prior knowledge of ɛ. Note that no classical randomized algorithm can reduce the error probability to significantly below ɛ t+1, even if ɛ is known. In this paper, we obtain lower bounds that show that the amplification achieved by these quantum algorithms is essentially optimal. We also present simple alternative algorithms that achieve the same bound as those in Grover [G05], and have some other desirable properties. We then study the best reduction in error that can be achieved by a tquery quantum algorithm, when the initial error ɛ is known to lie in an interval of the form [ℓ, u]. We generalize our basic algorithms and lower bounds, and obtain nearly tight bounds in this setting. 1
Speedup via Quantum Sampling
 Physical Review A
, 2008
"... The Markov Chain Monte Carlo method is at the heart of most fullypolynomial randomized approximation schemes for #Pcomplete problems such as estimating the permanent or the value of a polytope. It is therefore very natural and important to determine whether quantum computers can speedup classical ..."
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The Markov Chain Monte Carlo method is at the heart of most fullypolynomial randomized approximation schemes for #Pcomplete problems such as estimating the permanent or the value of a polytope. It is therefore very natural and important to determine whether quantum computers can speedup classical mixing processes based on Markov chains. To this end, we present a new quantum algorithm, making it possible to prepare a quantum sample, i.e., a coherent version of the stationary distribution of a reversible Markov chain. Our algorithm has a significantly better running time than that of a previous algorithm based on adiabatic state generation. We also show that our methods provide a speedup over a recently proposed method for obtaining ground states of (classical) Hamiltonians. In an upcoming article, we will show that they yield speedups of classical algorithms for approximately evaluating the permanent. 1
Fault Models for Quantum Mechanical Switching Networks
, 2005
"... This work justifies several quantum gate level fault models and discusses the causal error mechanisms thwarting correct function. A quantum adaptation of the classical test set generation technique known as constructing a fault table is given. This classical technique optimizes test plans to detect ..."
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This work justifies several quantum gate level fault models and discusses the causal error mechanisms thwarting correct function. A quantum adaptation of the classical test set generation technique known as constructing a fault table is given. This classical technique optimizes test plans to detect all the most common error types. This work therefore considers the set of predominate errors modeled by unwanted qubit rotations. In classical test, a fault table is constructed allowing the comparison between a circuit’s nominal response and a response perturbed by each separately considered error. It was found that isolating a correct circuit from a circuit containing any of the Pauli Fault rotations, requires applications of just two independent test vectors. This is related to the proven fact that a reversible system preserves the probability that additional information may be present. Thus, the probability of detection for an observable fault is related only to the probability of presence. A theorem that better connects classical ideas to quantum test set generation is presented. This leads directly to a relationship between the deterministic presence of a fault in the state vector observed with some probability and the probabilistic presence of a fault observed deterministically (Relating Time and Space Error Models).
The Phase Matrix
"... Abstract. Reducing the error of quantum algorithms is often achieved by applying a primitive called amplitude amplification. Its use leads in many instances to quantum algorithms that are quadratically faster than any classical algorithm. Amplitude amplification is controlled by choosing two complex ..."
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Abstract. Reducing the error of quantum algorithms is often achieved by applying a primitive called amplitude amplification. Its use leads in many instances to quantum algorithms that are quadratically faster than any classical algorithm. Amplitude amplification is controlled by choosing two complex numbers φs and φt of unit norm, called phase factors. If the phases are wellchosen, amplitude amplification reduces the error of quantum algorithms, if not, it may increase the error. We give an analysis of amplitude amplification with a emphasis on the influence of the phase factors on the error of quantum algorithms. We introduce a socalled phase matrix and use it to give a straightforward and novel analysis of amplitude amplification processes. We show that we may always pick identical phase factors φs = φt with argument in the range
Quantum Speedup for Approximating Partition Functions
, 2008
"... We achieve a quantum speedup of fully polynomial randomized approximation schemes (FPRAS) for estimating partition functions that combine simulated annealing with the MonteCarlo Markov Chain method and use nonadaptive cooling schedules. The improvement in time complexity is twofold: a quadratic r ..."
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We achieve a quantum speedup of fully polynomial randomized approximation schemes (FPRAS) for estimating partition functions that combine simulated annealing with the MonteCarlo Markov Chain method and use nonadaptive cooling schedules. The improvement in time complexity is twofold: a quadratic reduction with respect to the spectral gap of the underlying Markov chains and a quadratic reduction with respect to the parameter characterizing the desired accuracy of the estimate output by the FPRAS. Both reductions are intimately related and cannot be achieved separately. First, we use Grover’s fixed point search, quantum walks and phase estimation to efficiently prepare approximate coherent encodings of stationary distributions of the Markov chains. The speedup we obtain in this way is due to the quadratic relation between the spectral and phase gaps of classical and quantum walks. Second, we generalize the method of quantum counting, showing how to estimate expected values of quantum observables. Using this method instead of classical sampling, we obtain the speedup with respect to accuracy. 1
Quantum Searching amidst Uncertainty
, 2005
"... Consider a database most of whose entries are marked but the precise fraction of marked entries is not known. What is known is that the fraction of marked entries is 1 − ǫ, where ǫ is a random variable that is uniformly distributed in the range (0,ǫ0).The problem is to try to select a marked item fr ..."
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Consider a database most of whose entries are marked but the precise fraction of marked entries is not known. What is known is that the fraction of marked entries is 1 − ǫ, where ǫ is a random variable that is uniformly distributed in the range (0,ǫ0).The problem is to try to select a marked item from the database in a single query. If the algorithm selects a marked item, it succeeds, else if it selects an unmarked item, it makes an error. How low can we make the probability of error? The best possible classical algorithm can lower the probability of error to O ( ǫ2) 0. The best known quantum algorithms for this problem could also only lower the probability of error to O ( ǫ2) 0. Using a recently invented quantum search technique, this paper gives an algorithm that reduces the probability of error to O ( ǫ3) 0. The algorithm is asymptotically optimal. 1