We show that in a quantum adiabatic evolution, even though the adiabatic approximation is valid, ... more We show that in a quantum adiabatic evolution, even though the adiabatic approximation is valid, the total phase of the final state indicated by the adiabatic theorem may evidently differ from the actual total phase. This invalidates the application of the linearity and the adiabatic approximation simultaneously. Besides, based on this observation we point out the mistake in the traditional proof for the adiabatic theorem. This mistake is the root of the troubles that the adiabatic theorem has met. We also show that a similar mistake remains in some recent modifications of the traditional adiabatic theorem attempting to eliminate the troubles.
Grover's algorithm is one of the most important quantum algorithms, which performs the task of se... more Grover's algorithm is one of the most important quantum algorithms, which performs the task of searching an unsorted database without a priori probability. Recently the adiabatic evolution has been used to design and reproduce quantum algorithms, including Grover's algorithm. In this paper, we show that quantum search algorithm by adiabatic evolution has two properties that conventional quantum search algorithm doesn't have. Firstly, we show that in the initial state of the algorithm only the amplitude of the basis state corresponding to the solution affects the running time of the algorithm, while other amplitudes do not. Using this property, if we know a priori probability about the location of the solution before search, we can modify the adiabatic evolution to make the algorithm faster. Secondly, we show that by a factor for the initial and finial Hamiltonians we can reduce the running time of the algorithm arbitrarily. Especially, we can reduce the running time of adiabatic search algorithm to a constant time independent of the size of the database. The second property can be extended to other adiabatic algorithms.
More and more quantum algorithms have been designed for solving problems in machine learning, dat... more More and more quantum algorithms have been designed for solving problems in machine learning, database search and data analytics. An important problem then arises: how privacy can be protected when these algorithms are used on private data? For classical computing, the notion of differential privacy provides a very useful conceptual framework in which a great number of mechanisms that protect privacy by introducing certain noises into algorithms have been successfully developed. This paper defines a notion of differential privacy for quantum information processing. We carefully examine how the mechanisms using three important types of quantum noise, the amplitude/phase damping and depolarizing, can protect differential privacy. A composition theorem is proved that enables us to combine multiple privacy-preserving operations in quantum information processing.
The LU-LC conjecture is an important open problem concerning the structure of entanglement of sta... more The LU-LC conjecture is an important open problem concerning the structure of entanglement of states described in the stabilizer formalism. It states that two local unitary equivalent stabilizer states are also local Clifford equivalent. If this conjecture were true, the local equivalence of stabilizer states would be extremely easy to characterize. Unfortunately, however, based on the recent progress made by Gross and Van den Nest, we find that the conjecture is false.
Quantum Hoare Logic (QHL) was introduced in Ying's work to specify and reason about quantum p... more Quantum Hoare Logic (QHL) was introduced in Ying's work to specify and reason about quantum programs. In this paper, we implement a theorem prover for QHL based on Isabelle/HOL. By applying the theorem prover, verifying a quantum program against a specification is transformed equivalently into an order relation between matrices. Due to the limitation of Isabelle/HOL, the calculation of the order relation is solved by calling an outside oracle written in Python. To the best of our knowledge, this is the first theorem prover for quantum programs. To demonstrate its power, the correctness of two well-known quantum algorithms, i.e., Grover Quantum Search and Quantum Phase Estimation (the key step in Shor's quantum algorithm of factoring in polynomial time) are proved using the theorem prover. These are the first mechanized proofs for both of them.
The semiring-based constraint satisfaction problems (semiring CSPs), proposed by Bistarelli, Mont... more The semiring-based constraint satisfaction problems (semiring CSPs), proposed by Bistarelli, Montanari and Rossi [3], is a very general framework of soft constraints. In this paper we propose an abstraction scheme for soft constraints that uses semiring homomorphism. To find optimal solutions of the concrete problem, the idea is, first working in the abstract problem and finding its optimal solutions, then using them to solve the concrete problem. In particular, we show that a mapping preserves optimal solutions if and only if it is an order-reflecting semiring homomorphism. Moreover, for a semiring homomorphism α and a problem P over S, if t is optimal in α(P), then there is an optimal solutiont of P such thatt has the same value as t in α(P).
Quantum walks are very useful tools in designing quantum algorithms. Amplitude amplification is a... more Quantum walks are very useful tools in designing quantum algorithms. Amplitude amplification is a key technique to increase the success probability of a quantum-walk-based algorithm, and it is quadratically faster than classical probabilistic amplification. However, amplitude amplification only applies to quantum walks with one-shot hitting time, where no measurements except a final one are performed, and not to quantum walks with concurrent hitting time, where measurements happen or absorbing boundaries exist at each step. In this paper, we propose a procedure to modify quantum walks with concurrent hitting time by removing measurements from them. This procedure enables us to use amplitude amplification to design algorithms based on the modified quantum walks which are faster than those based on the original walks with a concurrent hitting time and more robust than those based on the corresponding walks with a one-shot hitting time.
We prove that any multiple-copy entanglement transformation [S. Bandyopadhyay, V. Roychowdhury, a... more We prove that any multiple-copy entanglement transformation [S. Bandyopadhyay, V. Roychowdhury, and U. Sen, Phys. Rev. A 65, 052315 (2002)] can be implemented by a suitable entanglementassisted local transformation [D. Jonathan and M. B. Plenio, Phys. Rev. Lett. 83, 3566 (1999)]. Furthermore, we show that the combination of multiple-copy entanglement transformation and the entanglement-assisted one is still equivalent to the pure entanglement-assisted one. The mathematical structure of multiple-copy entanglement transformations then is carefully investigated. Many interesting properties of multiple-copy entanglement transformations are presented, which exactly coincide with those satisfied by the entanglement-assisted ones. Most interestingly, we show that an arbitrarily large number of copies of state should be considered in multiple-copy entanglement transformations.
Usually, probabilistic automata and probabilistic grammars have crisp symbols as inputs, which ca... more Usually, probabilistic automata and probabilistic grammars have crisp symbols as inputs, which can be viewed as the formal models of computing with values. In this paper, we first introduce probabilistic automata and probabilistic grammars for computing with (some special) words in a probabilistic framework, where the words are interpreted as probabilistic distributions or possibility distributions over a set of crisp symbols. By probabilistic conditioning, we then establish a retraction principle from computing with words to computing with values for handling crisp inputs and a generalized extension principle from computing with words to computing with all words for handling arbitrary inputs. These principles show that computing with values and computing with all words can be respectively implemented by computing with some special words. To compare the transition probabilities of two near inputs, we also examine some analytical properties of the transition probability functions of generalized extensions. Moreover, the retractions and the generalized extensions are shown to be equivalence-preserving. Finally, we clarify some relationships among the retractions, the generalized extensions, and the extensions studied recently by Qiu and Wang.
Quantum adiabatic computation is a novel paradigm for the design of quantum algorithms, which is ... more Quantum adiabatic computation is a novel paradigm for the design of quantum algorithms, which is usually used to find the minimum of a classical function. In this paper, we show that if the initial hamiltonian of a quantum adiabatic evolution with a interpolation path is too simple, the minimal gap between the ground state and the first excited state of this quantum adiabatic evolution is an inverse exponential distance. Thus quantum adiabatic evolutions of this kind can't be used to design efficient quantum algorithms. Similarly, we show that a quantum adiabatic evolution with a simple final hamiltonian also has a long running time, which suggests that some functions can't be minimized efficiently by any quantum adiabatic evolution with a interpolation path.
Direction relations between extended spatial objects are important commonsense knowledge. Recentl... more Direction relations between extended spatial objects are important commonsense knowledge. Recently, Goyal and Egenhofer proposed a formal model, called Cardinal Direction Calculus (CDC), for representing direction relations between connected plane regions. CDC is perhaps the most expressive qualitative calculus for directional information, and has attracted increasing interest from areas such as artificial intelligence, geographical information science, and image retrieval. Given a network of CDC constraints, the consistency problem is deciding if the network is realizable by connected regions in the real plane. This paper provides a cubic algorithm for checking consistency of basic CDC constraint networks. As one byproduct, we also show that any consistent network of CDC constraints has a canonical realization in digital plane. The cubic algorithm can also been adapted to cope with disconnected regions, in which case the current best algorithm is of time complexity O(n 5).
We propose a series of quantum algorithms for computing a wide range of quantum entropies and dis... more We propose a series of quantum algorithms for computing a wide range of quantum entropies and distances, including the von Neumann entropy, quantum Rényi entropy, trace distance, and fidelity. The proposed algorithms significantly outperform the best known (and even quantum) ones in the low-rank case, some of which achieve exponential speedups. In particular, for Ndimensional quantum states of rank r, our proposed quantum algorithms for computing the von Neumann entropy, trace distance and fidelity within additive error ε have time complexity ofÕ r 2 /ε 2 ,Õ r 5 /ε 6 andÕ r 6.5 /ε 7.5 1 , respectively. In contrast, the known algorithms for the von Neumann entropy and trace distance require quantum time complexity of Ω(N) [AISW19,GL20,GHS21], and the best known one for fidelity requiresÕ r 21.5 /ε 23.5 [WZC + 21]. The key idea of our quantum algorithms is to extend block-encoding from unitary operators in previous work to quantum states (i.e., density operators). It is realized by developing several convenient techniques to manipulate quantum states and extract information from them. In particular, we introduce a novel technique for eigenvalue transformation of density operators and their (non-integer) positive powers, based on the powerful quantum singular value transformation (QSVT) [GSLW19]. The advantage of our techniques over the existing methods is that no restrictions on density operators are required; in sharp contrast, the previous methods usually require a lower bound of the minimal non-zero eigenvalue of density operators. In addition, we provide some techniques of independent interest for trace estimation, linear combinations, and eigenvalue threshold projectors of (subnormalized) density operators, which will be, we believe, useful in other quantum algorithms.
The quantum coupling of two given quantum states denotes the set of bipartite states whose margin... more The quantum coupling of two given quantum states denotes the set of bipartite states whose marginal states are these given two states. In this paper, we provide tight inequalities to describe the structure of quantum coupling. These inequalities directly imply that the trace distance between two quantum states cannot be determined by the quantum analog of the earth mover’s distance, thus ruling out the equality version of the quantum Kantorovich–Rubinstein theorem for trace distance even in the finite-dimensional case. In addition, we provide an inequality that can be regarded as a quantum generalization of the Kantorovich–Rubinstein theorem. Then, we generalize our inequalities and apply them to the three tripartite quantum marginal problems. Numerical tests with a three-qubit system show that our criteria are much stronger than the known criteria: the strong subadditivity of entropy and the monogamy of entanglement.
Markov decision process (MDP) offers a general framework for modelling sequential decision making... more Markov decision process (MDP) offers a general framework for modelling sequential decision making where outcomes are random. In particular, it serves as a mathematical framework for reinforcement learning. This paper introduces an extension of MDP, namely quantum MDP (qMDP), that can serve as a mathematical model of decision making about quantum systems. We develop dynamic programming algorithms for policy evaluation and finding optimal policies for qMDPs in the case of finite-horizon. The results obtained in this paper provide some useful mathematical tools for reinforcement learning techniques applied to the quantum world.
We show that in a quantum adiabatic evolution, even though the adiabatic approximation is valid, ... more We show that in a quantum adiabatic evolution, even though the adiabatic approximation is valid, the total phase of the final state indicated by the adiabatic theorem may evidently differ from the actual total phase. This invalidates the application of the linearity and the adiabatic approximation simultaneously. Besides, based on this observation we point out the mistake in the traditional proof for the adiabatic theorem. This mistake is the root of the troubles that the adiabatic theorem has met. We also show that a similar mistake remains in some recent modifications of the traditional adiabatic theorem attempting to eliminate the troubles.
Grover's algorithm is one of the most important quantum algorithms, which performs the task of se... more Grover's algorithm is one of the most important quantum algorithms, which performs the task of searching an unsorted database without a priori probability. Recently the adiabatic evolution has been used to design and reproduce quantum algorithms, including Grover's algorithm. In this paper, we show that quantum search algorithm by adiabatic evolution has two properties that conventional quantum search algorithm doesn't have. Firstly, we show that in the initial state of the algorithm only the amplitude of the basis state corresponding to the solution affects the running time of the algorithm, while other amplitudes do not. Using this property, if we know a priori probability about the location of the solution before search, we can modify the adiabatic evolution to make the algorithm faster. Secondly, we show that by a factor for the initial and finial Hamiltonians we can reduce the running time of the algorithm arbitrarily. Especially, we can reduce the running time of adiabatic search algorithm to a constant time independent of the size of the database. The second property can be extended to other adiabatic algorithms.
More and more quantum algorithms have been designed for solving problems in machine learning, dat... more More and more quantum algorithms have been designed for solving problems in machine learning, database search and data analytics. An important problem then arises: how privacy can be protected when these algorithms are used on private data? For classical computing, the notion of differential privacy provides a very useful conceptual framework in which a great number of mechanisms that protect privacy by introducing certain noises into algorithms have been successfully developed. This paper defines a notion of differential privacy for quantum information processing. We carefully examine how the mechanisms using three important types of quantum noise, the amplitude/phase damping and depolarizing, can protect differential privacy. A composition theorem is proved that enables us to combine multiple privacy-preserving operations in quantum information processing.
The LU-LC conjecture is an important open problem concerning the structure of entanglement of sta... more The LU-LC conjecture is an important open problem concerning the structure of entanglement of states described in the stabilizer formalism. It states that two local unitary equivalent stabilizer states are also local Clifford equivalent. If this conjecture were true, the local equivalence of stabilizer states would be extremely easy to characterize. Unfortunately, however, based on the recent progress made by Gross and Van den Nest, we find that the conjecture is false.
Quantum Hoare Logic (QHL) was introduced in Ying's work to specify and reason about quantum p... more Quantum Hoare Logic (QHL) was introduced in Ying's work to specify and reason about quantum programs. In this paper, we implement a theorem prover for QHL based on Isabelle/HOL. By applying the theorem prover, verifying a quantum program against a specification is transformed equivalently into an order relation between matrices. Due to the limitation of Isabelle/HOL, the calculation of the order relation is solved by calling an outside oracle written in Python. To the best of our knowledge, this is the first theorem prover for quantum programs. To demonstrate its power, the correctness of two well-known quantum algorithms, i.e., Grover Quantum Search and Quantum Phase Estimation (the key step in Shor's quantum algorithm of factoring in polynomial time) are proved using the theorem prover. These are the first mechanized proofs for both of them.
The semiring-based constraint satisfaction problems (semiring CSPs), proposed by Bistarelli, Mont... more The semiring-based constraint satisfaction problems (semiring CSPs), proposed by Bistarelli, Montanari and Rossi [3], is a very general framework of soft constraints. In this paper we propose an abstraction scheme for soft constraints that uses semiring homomorphism. To find optimal solutions of the concrete problem, the idea is, first working in the abstract problem and finding its optimal solutions, then using them to solve the concrete problem. In particular, we show that a mapping preserves optimal solutions if and only if it is an order-reflecting semiring homomorphism. Moreover, for a semiring homomorphism α and a problem P over S, if t is optimal in α(P), then there is an optimal solutiont of P such thatt has the same value as t in α(P).
Quantum walks are very useful tools in designing quantum algorithms. Amplitude amplification is a... more Quantum walks are very useful tools in designing quantum algorithms. Amplitude amplification is a key technique to increase the success probability of a quantum-walk-based algorithm, and it is quadratically faster than classical probabilistic amplification. However, amplitude amplification only applies to quantum walks with one-shot hitting time, where no measurements except a final one are performed, and not to quantum walks with concurrent hitting time, where measurements happen or absorbing boundaries exist at each step. In this paper, we propose a procedure to modify quantum walks with concurrent hitting time by removing measurements from them. This procedure enables us to use amplitude amplification to design algorithms based on the modified quantum walks which are faster than those based on the original walks with a concurrent hitting time and more robust than those based on the corresponding walks with a one-shot hitting time.
We prove that any multiple-copy entanglement transformation [S. Bandyopadhyay, V. Roychowdhury, a... more We prove that any multiple-copy entanglement transformation [S. Bandyopadhyay, V. Roychowdhury, and U. Sen, Phys. Rev. A 65, 052315 (2002)] can be implemented by a suitable entanglementassisted local transformation [D. Jonathan and M. B. Plenio, Phys. Rev. Lett. 83, 3566 (1999)]. Furthermore, we show that the combination of multiple-copy entanglement transformation and the entanglement-assisted one is still equivalent to the pure entanglement-assisted one. The mathematical structure of multiple-copy entanglement transformations then is carefully investigated. Many interesting properties of multiple-copy entanglement transformations are presented, which exactly coincide with those satisfied by the entanglement-assisted ones. Most interestingly, we show that an arbitrarily large number of copies of state should be considered in multiple-copy entanglement transformations.
Usually, probabilistic automata and probabilistic grammars have crisp symbols as inputs, which ca... more Usually, probabilistic automata and probabilistic grammars have crisp symbols as inputs, which can be viewed as the formal models of computing with values. In this paper, we first introduce probabilistic automata and probabilistic grammars for computing with (some special) words in a probabilistic framework, where the words are interpreted as probabilistic distributions or possibility distributions over a set of crisp symbols. By probabilistic conditioning, we then establish a retraction principle from computing with words to computing with values for handling crisp inputs and a generalized extension principle from computing with words to computing with all words for handling arbitrary inputs. These principles show that computing with values and computing with all words can be respectively implemented by computing with some special words. To compare the transition probabilities of two near inputs, we also examine some analytical properties of the transition probability functions of generalized extensions. Moreover, the retractions and the generalized extensions are shown to be equivalence-preserving. Finally, we clarify some relationships among the retractions, the generalized extensions, and the extensions studied recently by Qiu and Wang.
Quantum adiabatic computation is a novel paradigm for the design of quantum algorithms, which is ... more Quantum adiabatic computation is a novel paradigm for the design of quantum algorithms, which is usually used to find the minimum of a classical function. In this paper, we show that if the initial hamiltonian of a quantum adiabatic evolution with a interpolation path is too simple, the minimal gap between the ground state and the first excited state of this quantum adiabatic evolution is an inverse exponential distance. Thus quantum adiabatic evolutions of this kind can't be used to design efficient quantum algorithms. Similarly, we show that a quantum adiabatic evolution with a simple final hamiltonian also has a long running time, which suggests that some functions can't be minimized efficiently by any quantum adiabatic evolution with a interpolation path.
Direction relations between extended spatial objects are important commonsense knowledge. Recentl... more Direction relations between extended spatial objects are important commonsense knowledge. Recently, Goyal and Egenhofer proposed a formal model, called Cardinal Direction Calculus (CDC), for representing direction relations between connected plane regions. CDC is perhaps the most expressive qualitative calculus for directional information, and has attracted increasing interest from areas such as artificial intelligence, geographical information science, and image retrieval. Given a network of CDC constraints, the consistency problem is deciding if the network is realizable by connected regions in the real plane. This paper provides a cubic algorithm for checking consistency of basic CDC constraint networks. As one byproduct, we also show that any consistent network of CDC constraints has a canonical realization in digital plane. The cubic algorithm can also been adapted to cope with disconnected regions, in which case the current best algorithm is of time complexity O(n 5).
We propose a series of quantum algorithms for computing a wide range of quantum entropies and dis... more We propose a series of quantum algorithms for computing a wide range of quantum entropies and distances, including the von Neumann entropy, quantum Rényi entropy, trace distance, and fidelity. The proposed algorithms significantly outperform the best known (and even quantum) ones in the low-rank case, some of which achieve exponential speedups. In particular, for Ndimensional quantum states of rank r, our proposed quantum algorithms for computing the von Neumann entropy, trace distance and fidelity within additive error ε have time complexity ofÕ r 2 /ε 2 ,Õ r 5 /ε 6 andÕ r 6.5 /ε 7.5 1 , respectively. In contrast, the known algorithms for the von Neumann entropy and trace distance require quantum time complexity of Ω(N) [AISW19,GL20,GHS21], and the best known one for fidelity requiresÕ r 21.5 /ε 23.5 [WZC + 21]. The key idea of our quantum algorithms is to extend block-encoding from unitary operators in previous work to quantum states (i.e., density operators). It is realized by developing several convenient techniques to manipulate quantum states and extract information from them. In particular, we introduce a novel technique for eigenvalue transformation of density operators and their (non-integer) positive powers, based on the powerful quantum singular value transformation (QSVT) [GSLW19]. The advantage of our techniques over the existing methods is that no restrictions on density operators are required; in sharp contrast, the previous methods usually require a lower bound of the minimal non-zero eigenvalue of density operators. In addition, we provide some techniques of independent interest for trace estimation, linear combinations, and eigenvalue threshold projectors of (subnormalized) density operators, which will be, we believe, useful in other quantum algorithms.
The quantum coupling of two given quantum states denotes the set of bipartite states whose margin... more The quantum coupling of two given quantum states denotes the set of bipartite states whose marginal states are these given two states. In this paper, we provide tight inequalities to describe the structure of quantum coupling. These inequalities directly imply that the trace distance between two quantum states cannot be determined by the quantum analog of the earth mover’s distance, thus ruling out the equality version of the quantum Kantorovich–Rubinstein theorem for trace distance even in the finite-dimensional case. In addition, we provide an inequality that can be regarded as a quantum generalization of the Kantorovich–Rubinstein theorem. Then, we generalize our inequalities and apply them to the three tripartite quantum marginal problems. Numerical tests with a three-qubit system show that our criteria are much stronger than the known criteria: the strong subadditivity of entropy and the monogamy of entanglement.
Markov decision process (MDP) offers a general framework for modelling sequential decision making... more Markov decision process (MDP) offers a general framework for modelling sequential decision making where outcomes are random. In particular, it serves as a mathematical framework for reinforcement learning. This paper introduces an extension of MDP, namely quantum MDP (qMDP), that can serve as a mathematical model of decision making about quantum systems. We develop dynamic programming algorithms for policy evaluation and finding optimal policies for qMDPs in the case of finite-horizon. The results obtained in this paper provide some useful mathematical tools for reinforcement learning techniques applied to the quantum world.
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Papers by Mingsheng Ying