Simulating and post-processing using a generative adversarial network
A hybrid computing system comprising a quantum computer and a digital computer employs a digital computer to use machine learning methods for post-processing samples drawn from the quantum computer. Post-processing samples can include simulating samples drawn from the quantum computer. Machine learning methods such as generative adversarial networks (GANs) and conditional GANs are applied. Samples drawn from the quantum computer can be a target distribution. A generator of a GAN generates samples based on a noise prior distribution and a discriminator of a GAN measures the distance between the target distribution and a generative distribution. A generator parameter and a discriminator parameter are respectively minimized and maximized.
1. A method of computationally efficiently producing sample sets in a processor-based system, comprising:
initializing a generator parameter θ;
initializing a discriminator parameter ϕ;
drawing a noise sample z k from a noise prior distribution r(z);
for each respective noise sample z k drawn from the noise prior distribution r(z), drawing a generated sample x (m|k) from a generator g θ (xÅz);
drawing a target sample {circumflex over (x)} (k) from a target distribution h,J that was generated by a quantum processor for a set of biases h and a set of coupling strengths J, where k=1, . . . , K;
adjusting the generator parameter θ;
adjusting the discriminator parameter ϕ; and
determining whether the adjusted generator parameter θ and the adjusted discriminator parameter ϕ each meet respective optimization criteria.
2. The method of claim 1 , further comprising:
in response to a determination that at least one of the adjusted generator parameter θ or the adjusted discriminator parameter ϕ do not meet the respective optimization criteria, drawing a new noise sample z k from the noise prior distribution r(z).
3. The method of claim 1 , further comprising:
in a first iteration, receiving a set of biases and a set of coupling strength values before drawing the noise sample from the noise prior distribution r(z);
in response to a determination that at least one of the adjusted generator parameter θ or the adjusted discriminator parameter ϕ do not meet the respective optimization criteria, in a second iteration:
receiving a new set of biases and a new set of coupling strength values before drawing a new noise sample in the second iteration; and
drawing the new noise sample z k from the noise prior distribution r(z) in the second iteration.
4. The method of claim 1 wherein drawing a noise sample z k from a noise prior distribution r(z) includes drawing a noise sample z k from a noise prior distribution r(z) that includes K noise samples z k , where k=1, . . . , K.
5. The method of claim 4 further comprising generating a generated sample x for each noise sample z k drawn from noise prior distribution r(z) via a generator function g θ (x|z), wherein the generator function g θ (x|z) determines a pointwise probability q θ (x) of a generative distribution θ defined by:
q θ ( x )∫ z =g θ ( x|z )· r ( z ) dz
6. The method of claim 5 further comprising generating M generated samples x (m|k) via the generative distribution θ , each generated sample x (m|k) being a respective vector of qubit states defined on {−1,1} n , where m=1, . . . , M.
7. The method of claim 1 , further comprising:
conditioning at least one of the generator or the discriminator on multiple sets of biases and multiple sets of coupling strengths.
8. The method of claim 7 , further comprising:
employing a different set of biases and a different set of coupling strengths during each of a plurality of iterations.
9. The method of claim 1 , further comprising:
generating samples by a quantum processor for a set of biases and a set of coupling strengths.
10. A processor-based system to computationally efficiently producing sample sets, comprising:
at least one processor;
at least one nontransitory processor-readable medium communicatively coupled to the at least one processor and which stores processor executable instructions which, when executed by the at least one processor, cause the at least one processor to:
initialize a generator parameter θ;
initialize a discriminator parameter ϕ;
draw a noise sample z k from a noise prior distribution r(z);
for each respective noise sample z k drawn from the noise prior distribution r(z), draw a generated sample x (m|k) from a generator g θ (x|z);
draw a target sample {circumflex over (x)} (k) from a target distribution h,J that was generated by a quantum processor for a set of biases h and a set of coupling strengths J, where k=1, . . . , K;
adjust the generator parameter θ;
adjust the discriminator parameter ϕ; and
determine whether the adjusted generator parameter θ and the adjusted discriminator parameter ϕ each meet respective optimization criteria.
11. The processor-based system of claim 10 wherein, when executed, the processor executable instructions cause the processor further to:
in response to a determination that at least one of the adjusted generator parameter θ or the adjusted discriminator parameter ϕ do not meet the respective optimization criteria, draw a new noise sample z k from the noise prior distribution r(z).
12. The processor-based system of claim 10 wherein, when executed, the processor executable instructions cause the processor further to:
in a first iteration, receive a set of biases and a set of coupling strength values before drawing the noise sample from the noise prior distribution r(z);
in response to a determination that at least one of the adjusted generator parameter θ or the adjusted discriminator parameter ϕ do not meet the respective optimization criteria, in a second iteration:
receive a new set of biases and a new set of coupling strength values before drawing a new noise sample in the second iteration; and
draw the new noise sample z k from the noise prior distribution r(z) in the second iteration.
13. The processor-based system of claim 10 wherein to draw a noise sample z k from a noise prior distribution r(z), the at least one processor draws a noise sample z k from a noise prior distribution r(z) that includes K noise samples z k , where k=1, . . . K.
14. The processor-based system of claim 13 wherein a pointwise probability q θ (x) of a generative distribution θ is defined by:
q θ ( x )=∫ z g θ ( x|z )· r ( z ) dz
where g θ (x|z) is a generator function, and wherein, when executed, the processor executable instructions cause the processor further to: generate a generated sample x for each noise sample z k drawn from noise prior distribution r(z) via the generator function g θ (x|z).
15. The processor-based system of claim 14 wherein, when executed, the processor executable instructions cause the processor further to: generate M generated samples x (m|k) via the generative distribution θ can, each generated sample x (m|k) being a respective vector of qubit states defined on {−1,1} n , where m=1, . . . , M.
16. The processor-based system of claim 10 wherein to determine whether the adjusted generator parameter and the adjusted discriminator parameter are optimized the at least one processor determines if a change in a step size of the discriminator parameter is negligible.
17. The processor-based system of claims 16 wherein to determine if a change in a step size of the discriminator parameter ϕ is negligible the at least one processor determines if discriminator parameter does not change appreciably compared a previous iteration.
18. The processor-based system of claim 10 wherein, when executed, the processor executable instructions cause the processor further to:
condition at least one of the generator or the discriminator on additional information, such as multiple sets of biases and multiple sets of coupling strengths.
19. The processor-based system of claim 18 wherein, when executed, the processor executable instructions cause the processor further to:
employ a different set of biases and a different set of coupling strengths during each of a plurality of iterations.
20. The processor-based system of claim 10 wherein the at least one processor includes at least one digital processor and at least one quantum processor, and wherein when executed, the processor executable instructions cause the processor further to transmit instructions to the at least one quantum processor which cause the at least one quantum processor to generate samples for a set of biases and a set of coupling strengths.