A toolkit to find the optimal quantum query complexity and query optimal quantum algorithm of arbitrary Boolean functions.
Project description
QuantumQueryOptimizer
By R. Teal Witter and Michael T. Czekanski
A toolkit to find the optimal quantum query complexity and query optimal quantum algorithm of arbitrary Boolean functions.
Consider a function f that maps from D to E where D is a subset of bitstrings of length n and E is the set of single bit outputs. In the query model, an algorithm looks at the bits of the input string x in D as few times as possible before correctly determing f(x). Given f, our program finds the optimal query complexity of a quantum algorithm that evaluates f and a span program (i.e. quantum algorithm) that meets this query complexity by solving a semidefinite program (SDP).
There are two ways to run our program.
First, explicitly specify the sets D and E.
Second, create one function that generates the set D for arbitrary bitstring length n
and another function that generates the set E from D according to f.
(Note: We provide example functions in boolean_functions.py
.)
Installation
Install via pip with pip install quantumqueryoptimizer
.
Example 1  Explicit Construction
We consider the Boolean function OR
on input bitstrings of length 2.
The output is '1'
if any bit is 1 and '0'
otherwise.
In this example, we explicitly define both D
and E
.
Then we call our function qqo.runSDP
after loading the
our package quantum_query_optimizer
as qqo
.
import quantum_query_optimizer as qqo
D = ['00', '01', '10', '11']
E = ['0', '1', '1', '1']
qqo.runSDP(D=D, E=E)
The corresponding output should look similar to:
n: 2
D: ['00', '01', '10', '11']
E: ['0', '1', '1', '1']
Optimal Query Complexity: 1.414
Number of Iterations: 73
Run Time: 0.067 seconds
Example 2  Function Construction
We again consider OR
on bitstrings of length 2.
In this example, though, we define functions to generate
all bitstrings of length n and evaluate the function OR
on D.
Then we pass our functions into qqo.runSDPForN
and specify
for which sizes of bitstring n
we want to solve the SDP.
import quantum_query_optimizer as qqo
qqo.runSDPForN(getD=qqo.getDAll, getE=qqo.getEOR, n_end=2, n_start=2))
The corresponding output should look similar to:
n: 2
D: ['00', '01', '10', '11']
E: ['0', '1', '1', '1']
Optimal Query Complexity: 1.414
Number of Iterations: 73
Run Time: 0.058 seconds
(You can find both examples in examples.py
.)
Semidefinite Program Formulation
We use Ben Reichardt's formulation of the SDP for
optimal quantum query complexity (described in Theorem 6.2
)
and query optimal span program (Lemma 6.5
) in
Span programs and quantum query complexity:
The general adversary bound is nearly tight for every boolean function.
Alternating Direction Method
To solve Reichardt's SDP,
we use Zaiwen Wen, Donald Goldfarb, and Wotao Yin's
Algorithm 1
described in
Alternating direction augmented Lagrangian methods for semidefinite programming.
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