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One of the earliest cryptographic applications of quantum information was to create quantum digital cash that could not be counterfeited. In this paper, we describe a new type of quantum money: quantum coins, where all coins of the same denomination are represented by identical quantum states. We state desirable security properties such as anonymity and unforgeability and propose two candidate quantum coin schemes: one using black box operations, and another using blind quantum computation.
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In a quantum money scheme, a bank can issue money that users cannot counterfeit. Similar to bills of paper money, most quantum money schemes assign a unique serial number to each money state, thus potentially compromising the privacy of the users of quantum money. However in a quantum coins scheme, just like the traditional currency coin scheme, all the money states are exact copies of each other, providing a better level of privacy for the users. A quantum money scheme can be private, i.e., only the bank can verify the money states, or public, meaning anyone can verify. In this work, we propose a way to lift any private quantum coin scheme -- which is known to exist based on the existence of one-way functions, due to Ji, Liu, and Song (CRYPTO18) -- to a scheme that closely resembles a public quantum coin scheme. Verification of a new coin is done by comparing it to the coins the user already possesses, by using a projector on to the symmetric subspace. No public coin scheme was known prior to this work. It is also the first construction that is very close to a public quantum money scheme and is provably secure based on standard assumptions. The lifting technique when instantiated with the private quantum coins scheme, due to Mosca and Stebila 2010, gives rise to the first construction that is very close to an inefficient unconditionally secure public quantum money scheme.
The dimensionality of the internal coin space of discrete-time quantum walks has a strong impact on the complexity and richness of the dynamics of quantum walkers. While two-dimensional coin operators are sufficient to define a certain range of dynamics on complex graphs, higher dimensional coins are necessary to unleash the full potential of discrete-time quantum walks. In this work we present an experimental realization of a discrete-time quantum walk on a line graph that, instead of two-dimensional, exhibits a four-dimensional coin space. Making use of the extra degree of freedom we observe multiple ballistic propagation speeds specific to higher dimensional coin operators. By implementing a scalable technique, we demonstrate quantum walks on circles of various sizes, as well as on an example of a Husimi cactus graph. The quantum walks are realized via time-multiplexing in a Michelson interferometer loop architecture, employing as the coin degrees of freedom the polarization and the traveling direction of the pulses in the loop. Our theoretical analysis shows that the platform supports implementations of quantum walks with arbitrary $4 times 4$ unitary coin operations, and usual quantum walks on a line with various periodic and twisted boundary conditions.
The analysis of a physical problem simplifies considerably when one uses a suitable coordinate system. We apply this approach to the discrete-time quantum walks with coins given by $2j+1$-dimensional Wigner rotation matrices (Wigner walks), a model which was introduced in T. Miyazaki et al., Phys. Rev. A 76, 012332 (2007). First, we show that from the three parameters of the coin operator only one is physically relevant for the limit density of the Wigner walk. Next, we construct a suitable basis of the coin space in which the limit density of the Wigner walk acquires a much simpler form. This allows us to identify various dynamical regimes which are otherwise hidden in the standard basis description. As an example, we show that it is possible to find an initial state which reduces the number of peaks in the probability distribution from generic $2j+1$ to a single one. Moreover, the models with integer $j$ lead to the trapping effect. The derived formula for the trapping probability reveals that it can be highly asymmetric and it deviates from purely exponential decay. Explicit results are given up to the dimension five.
By repeated trials, one can determine the fairness of a classical coin with a confidence which grows with the number of trials. A quantum coin can be in a superposition of heads and tails and its state is most generally a density matrix. Given a string of qubits representing a series of trials, one can measure them individually and determine the state with a certain confidence. We show that there is an improved strategy which measures the qubits after entangling them, which leads to a greater confidence. This strategy is demonstrated on the simulation facility of IBM quantum computers.
We study one-dimensional quantum walk with four internal degrees of freedom resulted from two entangled qubits. We will demonstrate that the entanglement between the qubits and its corresponding coin operator enable one to steer the walkers state from a classical to standard quantum-walk behavior, and a novel one. Additionally, we report on self-trapped behavior and perfect transfer with highest velocity for the walker. We also show that symmetry of probability density distribution, the most probable place to find the walker and evolution of the entropy are subject to initial entanglement between the qubits. In fact, we confirm that if the two qubits are separable (zero entanglement), entropy becomes minimum whereas its maximization happens if the two qubits are initially maximally entangled. We will make contrast between cases where the entangled qubits are affected by coin operator identically or else, and show considerably different deviation in walkers behavior and its properties.
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