Merge branch 'master' of https://github.com/rigetticomputing/grove in…

…to bernstein-vazirani

docs/index.rst

@@ -28,6 +28,7 @@ of which has its own self-contained documentation.
    phaseestimation
    grover
    bernstein_vazirani
+   simon
    deutsch_jozsa
    arbitrary_state
 

docs/simon.rst

@@ -0,0 +1,79 @@
+Simon's Algorithm
+=================
+
+Overview
+--------
+
+This module emulates Simon's Algorithm.
+
+Simon's problem is summarized as follows.
+A function \\(f :\\{0,1\\}^n\\rightarrow\\{0,1\\}^n\\) is promised to be either
+one-to-one, or two-to-one with some nonzero \\(n\\)-bit mask \\(s\\).
+The latter condition means that for any two different \\(n\\)-bit numbers
+\\(x\\) and \\(y\\), \\(f(x)=f(y)\\) if and only if \\(x\\oplus y = s\\). The problem
+then is to determine whether \\(f\\) is one-to-one or two-to-one, and, if
+the latter, what the mask \\(s\\) is, in as few queries to \\(f\\) as possible.
+
+The problem statement and algorithm can be explored further, at a high level,
+in reference [1]_. The implementation of the algorithm in this module,
+however, follows [2]_.
+
+
+Algorithm and Details
+---------------------
+This algorithm involves a quantum component and a classical component.
+The quantum part follows similarly to other blackbox oracle algorithms.
+First, assume a blackbox oracle \\(U_f\\) is available with the property
+$$U_f\\vert x\\rangle\\vert y\\rangle = \\vert x\\rangle\\vert y\\oplus f(x)\\rangle$$
+
+where the top \\(n\\) qubits \\(\\vert x \\rangle\\) are the input,
+and the bottom \\(n\\) qubits \\(\\vert y \\rangle\\) are called ancilla qubits.
+
+The input qubits are prepared with the ancilla qubits into the state
+$$(H^{\\otimes n} \\otimes I^{\\otimes n})\\vert 0\\rangle^{\\otimes n}\\vert 0\\rangle^{\\otimes n} = \\vert +\\rangle^{\\otimes n}\\vert 0\\rangle^{\\otimes n}$$
+and sent through a blackbox gate \\(U_f\\). Then, the Hadamard-Walsh transform
+\\(H^{\\otimes n}\\) is applied to the \\(n\\) input qubits, resulting in the state given by
+ $$(H^{\\otimes n} \\otimes I^{\\otimes n})U_f\\vert +\\rangle^{\\otimes n}\\vert 0\\rangle^{\\otimes n}$$
+
+It turns out the resulting \\(n\\) input qubits are in a uniform random state
+over the space killed by (modulo \\(2\\), bitwise) dot product with \\(s\\).
+This covers the one-to-one case as well, if one considers it to be the degenerate \\(s=0\\) case.
+
+Suppose we then measured the \\(n\\) input qubits, calling the bitstring output \\(y\\).
+The above property then requires \\(s\\cdot y = 0\\). The space of \\(y\\) that satisfies this is \\(n-1\\) dimensional.
+By running this quantum subroutine several times, \\(n-1\\) nonzero linearly independent bitstrings \\(y_i\\), \\(i = 0, \\ldots, n-2\\), can be found, each orthogonal to \\(s\\).
+
+This gives a system of \\(n-1\\) equations, with \\(n\\) unknowns for finding \\(s\\).
+One final nonzero bitstring \\(y^{\\prime}\\) can be classically found that is linearly independent to the other \\(y_i\\), but with the property that \\(s\\cdot y^{\\prime} = 1\\).
+The combination of \\(y^{\\prime}\\) and the \\(y_i\\) give a system of \\(n\\) independent equations that can then be solved for \\(s\\).
+
+By using a clever implementation of Gaussian Elimination and Back Substitution
+for mod-2 equations, as outlined in Reference [2]_,
+\\(s\\) can be found relatively quickly. By then
+sending separate input states \\(\\vert 0\\rangle\\)
+and \\(\\vert s\\rangle\\) through the blackbox \\(U_f\\),
+we can find whether or not \\(f(0) = f(s)\\) (in fact,
+any pair \\(\\vert x\\rangle\\) and \\(\\vert x\\oplus s\\rangle\\) will do as well). If so, we conclude \\(f\\) is
+two-to-one with mask \\(s\\); otherwise, \\(f\\) is one-to-one.
+
+Overall, this algorithm can be solved in \\(O(n^3)\\), i.e., polynomial, time,
+whereas the best classical algorithm requires exponential time.
+
+
+Source Code Docs
+----------------
+
+Here you can find documentation for the different submodules in phaseestimation.
+
+grove.simon.simon
+~~~~~~~~~~~~~~~~~
+
+.. automodule:: grove.simon.simon
+    :members:
+    :undoc-members:
+    :show-inheritance:
+
+.. rubric:: References
+
+.. [1] http://pages.cs.wisc.edu/~dieter/Courses/2010f-CS880/Scribes/05/lecture05.pdf
+.. [2] http://lapastillaroja.net/wp-content/uploads/2016/09/Intro_to_QC_Vol_1_Loceff.pdf

docs/vqe.rst

@@ -395,7 +395,7 @@ Here are some links to get you started:
 
 - `A Variational Eigenvalue Solver on a Quantum Processor <https://arxiv.org/abs/1304.3061>`_
 
-- `The Theory of Variational Hybrid Quantum-Classical Algorithms <https://arxiv.org/abs/1304.3061>`_
+- `The Theory of Variational Hybrid Quantum-Classical Algorithms <https://arxiv.org/abs/1509.04279>`_
 
 - `Hybrid Quantum-Classical Approach to Correlated Materials <https://arxiv.org/abs/1510.03859>`_
 

grove/simon/__init__.py

@@ -0,0 +1 @@
+# Simon's algorithm init