By virtue of the new technique of performing integration over Dirac＇s ket-bra operators, we ex- plore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, ...By virtue of the new technique of performing integration over Dirac＇s ket-bra operators, we ex- plore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel- Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We cannot only find some new quantum mechanical unitary operators which correspond to the known optical transformations, de- riving a new theorem for calculating quantum tomogram of density operators, but also can reveal some new classical optical transformations. For examples, we find the generalized Fresnel opera- tor （GFO） to correspond to the generalized Fresnel transform （GFT） in classical optics. We derive GFO＇s normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in the context of quantum optics. Especially, the introduction of quantum mechanical entangled state representations opens up a new area in finding new classical optical transformations. The complex wavelet transform and the condition of mother wavelet are studied in the context of quantum op- tics too. Throughout our discussions, the coherent state, the entangled state representation of the two-mode squeezing operators and the technique of integration within an ordered product （IWOP） of operators are fully used. All these have confirmed Dirac＇s assertion： ＂...for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory＂.展开更多
A quantum circuit is a computational unit that transforms an input quantum state to an output state.A natural way to reason about its behavior is to compute explicitly the unitary matrix implemented by it.However,when...A quantum circuit is a computational unit that transforms an input quantum state to an output state.A natural way to reason about its behavior is to compute explicitly the unitary matrix implemented by it.However,when the number of qubits increases,the matrix dimension grows exponentially and the computation becomes intractable.In this paper,we propose a symbolic approach to reasoning about quantum circuits.It is based on a small set of laws involving some basic manipulations on vectors and matrices.This symbolic reasoning scales better than the explicit one and is well suited to be automated in Coq,as demonstrated with some typical examples.展开更多
文摘By virtue of the new technique of performing integration over Dirac＇s ket-bra operators, we ex- plore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel- Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We cannot only find some new quantum mechanical unitary operators which correspond to the known optical transformations, de- riving a new theorem for calculating quantum tomogram of density operators, but also can reveal some new classical optical transformations. For examples, we find the generalized Fresnel opera- tor （GFO） to correspond to the generalized Fresnel transform （GFT） in classical optics. We derive GFO＇s normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in the context of quantum optics. Especially, the introduction of quantum mechanical entangled state representations opens up a new area in finding new classical optical transformations. The complex wavelet transform and the condition of mother wavelet are studied in the context of quantum op- tics too. Throughout our discussions, the coherent state, the entangled state representation of the two-mode squeezing operators and the technique of integration within an ordered product （IWOP） of operators are fully used. All these have confirmed Dirac＇s assertion： ＂...for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory＂.
基金supported by the National Natural Science Foundation of China under Grant Nos.61832015 and 62072176the Research Funds of Happiness Flower East China Normal University under Grant No.2020ECNU-XFZH005+1 种基金the Inria-CAS Joint Project Quasar.Yuan Feng was partially supported by the National Key Research and Development Program of China under Grant No.2018YFA0306704the Australian Research Council under Grant No.DP180100691.
文摘A quantum circuit is a computational unit that transforms an input quantum state to an output state.A natural way to reason about its behavior is to compute explicitly the unitary matrix implemented by it.However,when the number of qubits increases,the matrix dimension grows exponentially and the computation becomes intractable.In this paper,we propose a symbolic approach to reasoning about quantum circuits.It is based on a small set of laws involving some basic manipulations on vectors and matrices.This symbolic reasoning scales better than the explicit one and is well suited to be automated in Coq,as demonstrated with some typical examples.
