Boltzmann machines are trained using an objective function that is evaluated by sampling quantum states that approximate a Gibbs state. Classical processing is used to produce the objective function, and the approximate Gibbs state is based on weights and biases that are refined using the sample results. In some examples, amplitude estimation is used. A combined classical/quantum computer produces suitable weights and biases for classification of shapes and other applications.
Please help me write a proper abstract based on the patent claims. CLAIM: 1.-15. (canceled) 16. A method of efficiently training a Boltzmann machine, comprising: with a classical computer, receiving a specification of a Boltzmann machine, an objective function, and associated training data, and in a quantum computer, determining at least one gradient of the objective function; based on the at least one gradient of the objective function, specifying at least one visible bias, at least one hidden bias or at least one weight of the Boltzmann machine so as to produce a trained Boltzmann machine. 17. The method of claim 16, further comprising: in the quantum computer, preparing a plurality of qubits so as to represent a Gibbs distribution; and producing the at least one gradient of the objective function by sampling the states of each of the plurality of qubits. 18. The method of claim 16, wherein the objective function is a sum of an average log-likelihood of the training data and a regularization function. 19. The method of claim 16, further comprising, in the quantum computer, producing a quantum state associated with model values, and in the classical computer, establishing the model values based on sampling of the quantum state. 20. The method of claim 19, wherein the model is revised using gradient ascent. 21. The method of claim 16, further comprising, in the quantum computer, producing a quantum state associated with data values, and in the classical computer, revising the model values based on sampling of the quantum state. 22. The method of claim 21, wherein the quantum state associated with the data values is produced based on a mean-field approximation to a Gibbs distribution. 23. A method, comprising: preparing at least one quantum state in a quantum computer to approximate a Gibbs state and sampling the at least one quantum state; and estimating in a classical computer, gradients of an objective function based on the sampling of the at least one quantum state. 24. The method of claim 23, further comprising defining weights or biases for the Boltzmann machine based on the estimated values. 25. The method of claim 24, further comprising processing a data example based on the defined weights and biases of the Boltzmann machine. 26. The method of claim 25, wherein the data example is associated with an image, a shape, speech, a text, an audio recording, a video recording, or a quantum state. 27. The method of claim 23, further comprising updating weights or biases associated with the Boltzmann machine based on gradients of at least one of Boltzmann machine weights, hidden biases, and visible biases. 28. The method of claim 23, wherein the objective function is associated with a sum of an average log-likelihood of the training data and a regularization function. 29. The method of claim 23, wherein the Boltzmann machine is a deep restricted Boltzmann machine having two or more layers. 30. The method of claim 23, further comprising; in the quantum computer, preparing a qubit string storing energy values associated with a plurality of hidden weights and biases; adding a qubit to the prepared qubit string and applying a rotation operator; applying amplitude estimation to the prepared qubit string with the added qubit; and in the classical computer, determining a gradient of an objective function based on the amplitude estimation. 31. The method of claim 23, wherein the preparing at least one state in the quantum computer comprises preparing a state associated with model weights, and hidden and variable biases. 32. The method of claim 31, further comprising computing a mean-field partition function, wherein the at least one state is based in part on the mean-field partition function. 33. The method of claim 23, wherein the preparing at least one state in the quantum computer comprises preparing a state associated with model weights, and hidden and variable biases, with visible unit fixed so as to correspond to a selected training vector. 34. The method of claim 33, further comprising computing a mean-field partition function, wherein the at least one state is based in part on the mean-field partition function associated with the selected training vector. 35. A method, comprising: in a classical computer, receiving a training data set, a specification of a deep Boltzmann machine, and an objective function; in a quantum computer, establishing a mean-field state based on the mean-field approximation so as to approximate a Gibbs state; and in the classical computer, estimating a gradient of the objective function based on sampling the mean-field state, and determining hidden biases and weights of the deep Boltzmann machine; and processing an input data example associated with an image, a shape, speech data, a text, an audio recording, a video or a quantum state output by a quantum device such as a quantum computer or simulator based on the Boltzmann machine.