Create simon_algorithm.py

chapter08/cirq/simon_algorithm.py

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+# Used for classical post-processing:
+from collections import Counter
+import numpy as np
+import scipy as sp
+import cirq
+
+"""Demonstrates Simon's algorithm.
+Simon's Algorithm solves the following problem:
+Given a function  f:{0,1}^n -> {0,1}^n, such that for some s ∈ {0,1}^n,
+f(x) = f(y) iff  x ⨁ y ∈ {0^n, s},
+find the n-bit string s.
+A classical algorithm requires O(2^n/2) queries to find s, while Simon’s
+algorithm needs only O(n) quantum queries.
+=== REFERENCE ===
+D. R. Simon. On the power of quantum cryptography. In35th FOCS, pages 116–123,
+Santa Fe,New Mexico, 1994. IEEE Computer Society Press.
+=== EXAMPLE OUTPUT ===
+Secret string = [1, 0, 0, 1, 0, 0]
+Circuit:
+                ┌──────┐   ┌───────────┐
+(0, 0): ────H────@──────────@─────@──────H───M('result')───
+                 │          │     │          │
+(1, 0): ────H────┼@─────────┼─────┼──────H───M─────────────
+                 ││         │     │          │
+(2, 0): ────H────┼┼@────────┼─────┼──────H───M─────────────
+                 │││        │     │          │
+(3, 0): ────H────┼┼┼@───────┼─────┼──────H───M─────────────
+                 ││││       │     │          │
+(4, 0): ────H────┼┼┼┼@──────┼─────┼──────H───M─────────────
+                 │││││      │     │          │
+(5, 0): ────H────┼┼┼┼┼@─────┼─────┼──────H───M─────────────
+                 ││││││     │     │
+(6, 0): ─────────X┼┼┼┼┼─────X─────┼───×────────────────────
+                  │││││           │   │
+(7, 0): ──────────X┼┼┼┼───────────┼───┼────────────────────
+                   ││││           │   │
+(8, 0): ───────────X┼┼┼───────────┼───┼────────────────────
+                    │││           │   │
+(9, 0): ────────────X┼┼───────────X───×────────────────────
+                     ││
+(10, 0): ────────────X┼────────────────────────────────────
+                      │
+(11, 0): ─────────────X────────────────────────────────────
+                └──────┘   └───────────┘
+Most common Simon Algorithm answer is: ('[1 0 0 1 0 0]', 100)
+***If the input string is s=0^n, no significant answer can be
+distinguished (since the null-space of the system of equations
+provided by the measurements gives a random vector). This will
+lead to low frequency count in output string.
+"""
+
+
+def main(qubit_count=3):
+
+    data = []  # we'll store here the results
+
+    # define a secret string:
+    secret_string = np.random.randint(2, size=qubit_count)
+
+    print(f'Secret string = {secret_string}')
+
+    n_samples = 100
+    for _ in range(n_samples):
+        flag = False  # check if we have a linearly independent set of measures
+        while not flag:
+            # Choose qubits to use.
+            input_qubits = [cirq.GridQubit(i, 0) for i in range(qubit_count)]  # input x
+            output_qubits = [
+                cirq.GridQubit(i + qubit_count, 0) for i in range(qubit_count)
+            ]  # output f(x)
+
+            # Pick coefficients for the oracle and create a circuit to query it.
+            oracle = make_oracle(input_qubits, output_qubits, secret_string)
+
+            # Embed oracle into special quantum circuit querying it exactly once
+            circuit = make_simon_circuit(input_qubits, output_qubits, oracle)
+
+            # Sample from the circuit a n-1 times (n = qubit_count).
+            simulator = cirq.Simulator()
+            results = [
+                simulator.run(circuit).measurements['result'][0] for _ in range(qubit_count - 1)
+            ]
+
+            # Classical Post-Processing:
+            flag = post_processing(data, results)
+
+    freqs = Counter(data)
+    print('Circuit:')
+    print(circuit)
+    print(f'Most common answer was : {freqs.most_common(1)[0]}')
+
+
+def make_oracle(input_qubits, output_qubits, secret_string):
+    """Gates implementing the function f(a) = f(b) iff a ⨁ b = s"""
+    # Copy contents to output qubits:
+    for control_qubit, target_qubit in zip(input_qubits, output_qubits):
+        yield cirq.CNOT(control_qubit, target_qubit)
+
+    # Create mapping:
+    if sum(secret_string):  # check if the secret string is non-zero
+        # Find significant bit of secret string (first non-zero bit)
+        significant = list(secret_string).index(1)
+
+        # Add secret string to input according to the significant bit:
+        for j in range(len(secret_string)):
+            if secret_string[j] > 0:
+                yield cirq.CNOT(input_qubits[significant], output_qubits[j])
+    # Apply a random permutation:
+    pos = [
+        0,
+        len(secret_string) - 1,
+    ]  # Swap some qubits to define oracle. We choose first and last:
+    yield cirq.SWAP(output_qubits[pos[0]], output_qubits[pos[1]])
+
+
+def make_simon_circuit(input_qubits, output_qubits, oracle):
+    """Solves for the secret period s of a 2-to-1 function such that
+    f(x) = f(y) iff x ⨁ y = s
+    """
+
+    c = cirq.Circuit()
+
+    # Initialize qubits.
+    c.append(
+        [
+            cirq.H.on_each(*input_qubits),
+        ]
+    )
+
+    # Query oracle.
+    c.append(oracle)
+
+    # Measure in X basis.
+    c.append([cirq.H.on_each(*input_qubits), cirq.measure(*input_qubits, key='result')])
+
+    return c
+
+
+def post_processing(data, results):
+    """Solves a system of equations with modulo 2 numbers"""
+    sing_values = sp.linalg.svdvals(results)
+    tolerance = 1e-5
+    if sum(sing_values < tolerance) == 0:  # check if measurements are linearly dependent
+        flag = True
+        null_space = sp.linalg.null_space(results).T[0]
+        solution = np.around(null_space, 3)  # chop very small values
+        minval = abs(min(solution[np.nonzero(solution)], key=abs))
+        solution = (solution / minval % 2).astype(int)  # renormalize vector mod 2
+        data.append(str(solution))
+        return flag
+
+
+if __name__ == '__main__':
+    main()