Quantum methods for Bayesian inference represent prior or current posterior distributions with a series of qubits. A rotation gate defined by a rotation angle based on the prior or current posterior is applied to a selected qubit of the series. The selected qubit is measured, and if measurement is successful, the state of the series of qubits represents a posterior or updated posterior. If the measurement is unsuccessful, the representation of the prior or current posterior in the series of qubits, the rotation operation, and the measurement operations are repeated until success. A sinc2 based model distribution is obtained using a quantum Fourier transform (QFT), and, in some cases, a QFT is also used to implement convolution in a filtering operation for inference with time-dependent systems.
Please help me write a proper abstract based on the patent claims. CLAIM: 1.-13. (canceled) 14. A method, comprising: (a) preparing a quantum state corresponding to a current posterior; (b) transforming the quantum state so as to produce a qubit string corresponding to a likelihood associated with the prior, the qubit string including at least one auxiliary qubit; (c) applying a rotation operation to the qubit string so that a state of the at least one auxiliary qubit is a superposition of a first state and a second state; (d) measuring the at least one auxiliary qubit, and if the measurement corresponds to the first state, determining a Bayesian update based on the qubit string; and (e) outputting a classical model for the posterior quantum state. 15. The method of claim 14, wherein the model for the posterior distribution is a function of the mean and/or the covariance matrix associated with the posterior distribution. 16. The method of claim 14, wherein the Bayesian update corresponds to a model. 17. The method of claim 14, further comprising repeating steps (b)-(d) for a set of measured data. 18. The method of claim 14, wherein the state of the qubit string if the measurement corresponds to the first state is Σ x  p  ( x )  p  ( E | x )   x ) Σ x  p  ( x )  p  ( e | x ) , wherein P(x) is a probability of a model x, and P(E|x) is a likelihood function based on measured data E. 19. The method of claim 14, wherein the posterior model corresponds to a Gaussian distribution. 20. The method of claim 14, wherein the posterior model corresponds to a sinc function. 21. The method of claim 20, wherein the sinc function corresponds to sin 2  ( π2 k  x / 2 n ) 2 k + n  sin 2  ( π   x / 2 n ) . 22. The method of claim 21, further comprising preparing a quantum state corresponding to the sinc function by: preparing a state  ψ 〉 = 1 2 k  ∑ j = 0 2 k - 1    j 〉 using k Hadamard gates for an integer k>0; applying a Pauli operator Z=P(½) to a least significant bit in |ψ; and applying a quantum Fourier transform. 23. The method of claim 15, further comprising updating the model of the posterior distribution by: using amplitude estimation to obtain a probability of successful quantum rejection sampling and a probability of obtaining a predetermined output in response to a Hadamard test; determining an updated mean or covariance matrix based on the probabilities. 24. The method of claim 14, wherein the Bayesian update corresponds to a mean or covariance matrix associated with the prior model, and further comprising: obtaining an estimate of at least a portion of a utility function using quantum rejection sampling; obtaining an estimate of a gradient of the utility function using a classical computer; and determining at least one subsequent measurement based on the estimate of the gradient. 25. The method of claim 23, wherein the loss function is a quadratic loss function, and at least one subsequent measurement is selected to reduce the expectation value of the quadratic loss function. 26. The method of claim 22, further comprising: (i) preparing the qubit register to represent the current posterior by: preparing a state  ψ 〉 = 1 2 k  ∑ j = 0 2 k - 1    j 〉 using k Hadamard gates for an integer k>0; applying a Pauli operator Z=P(½) to a least significant bit in |ψ; and applying a quantum Fourier transform; (ii) if the measurement corresponds to the first state, determining the Bayesian update based on the qubit string, wherein the quantum state corresponds to a prior model; and (iii) if the measurement corresponds to a second state, repeating (i) until the measurement corresponds to the first state, wherein the resultant quantum state corresponds to an updated posterior. 27. A method, comprising: (a) preparing a qubit register so as to represent a current posterior; (b) applying a quantum Fourier transform to the qubit register; (c) processing the Fourier-transformed qubit register to obtain a quantum state corresponding to a product with a Fourier transform of a predetermined distribution; (d) applying an inverse Fourier transform to the product and measuring a selected qubit of the Fourier transformed product; and (e) wherein if the measurement circuit indicates that the selected qubit is in a first state based on the measurement, representing a convolution of the current posterior and the predetermined distribution with the measured Fourier transformed product; (f) processing the resultant quantum state to obtain an updated current posterior. 28. The method of claim 27, wherein the predetermined distribution is a Gaussian distribution. 29. The method of claim 27, wherein the updated current posterior is obtained by quantum rejection sampling. 30. The method of claim 28, wherein if the measurement circuit indicates a second state of the selected qubit, repeating (a)-(e) until the measurement circuit indicates the first state, and processing the measured Fourier transformed product corresponding the measurement of the first state to obtain an updated current posterior. 31. A quantum computer, comprising: a qubit register situated to store a quantum state corresponding an input posterior distribution; a measurement circuit coupled to a selected qubit of the qubit register; a rotation gate coupled so as to rotate a state of the selected qubit through an angle proportional to the posterior, wherein if the measurement circuit reports that the selected qubit is in a first state when the qubit register corresponds to an updated posterior. 32. The quantum computer of claim 30, further comprising Hadamard gates, a Pauli gate, and gates corresponding to a quantum Fourier transform arranged to produce an estimate of the input posterior. 33. The quantum computer of claim 31, wherein the angle of rotation is proportional to a ratio of the posterior to a constant selected to be greater than or equal to the posterior and less than or equal to one.