[Examples] Add example to compute the expectation values

Signed-off-by: Shubhanshu Saxena <shubhanshu.e01@gmail.com>

examples/expectation_values/README.md

@@ -0,0 +1,28 @@
+## Benchmarking how far can we go simulating on classical computers
+
+**Problem:** How about computing exact expectation values in the form `<Ψ| Op |Ψ>`?
+
+### Instructions to run
+
+This script tries to simulate a 3-qubit system (011) and applies three different gates on the three qubits. Then we take `Op` as the Hadamard gate on the first qubit.
+
+```sh
+python examples/benchmark/speed_test.py
+```
+
+Parameters:
+
+1. `--sim-type <tensor | vector> (-s)`: Select the simulation type for the system. Default: vector
+
+### Explanation
+
+In quantum mechanics, the expectation value of any operator for any normalized vector `Ψ` is given by `<Ψ| Op |Ψ>`. Ref: [Expectation value](<https://en.wikipedia.org/wiki/Expectation_value_(quantum_mechanics)>)
+
+So, first we need to normalize the state of the quantum system (tensor or vector states).
+
+To measure the expectation values of the `<Ψ| Op |Ψ>`. We need to prepare `Op` for the 3-qubit system here where the other two qubits can be applied a quantum wire or Identity matrix. The given expression can be understood as two parts:
+
+1. `<Ψ|` as the hermitian conjugate of `|Ψ>` -> complex conjugate of transpose of `|Ψ>`
+2. `Op |Ψ>` as the `Op` multiplied with `psi` -> applying the operator on `|Ψ>` normally
+
+Once we get these two, we take the inner product of these two matrices we get from the above steps to get the expectation value.

examples/expectation_values/expectation_values.py

@@ -0,0 +1,66 @@
+import qcir_sim, argparse
+import numpy as np
+
+
+def parse_args():
+    parser = argparse.ArgumentParser(description="Benchmarking the simulation")
+    parser.add_argument(
+        "--sim-type",
+        "-s",
+        choices=["tensor", "vector"],
+        default="vector",
+        help="Type of simulation",
+    )
+    return parser.parse_args()
+
+
+def get_random_state():
+    choice = np.random.choice([0, 1])
+    if choice == 0:
+        return qcir_sim.zero()
+    return qcir_sim.one()
+
+
+def vector_expectation_value(sim: qcir_sim.VectorSimulation, gate: str, gate_idx: int):
+    psi = sim.vector_state
+    psi /= np.linalg.norm(psi)
+    Op = sim._prepare_gate(gate, gate_idx)
+
+    op_mul_psi = Op @ psi
+    bra_notation = np.conjugate(np.transpose(psi))
+
+    expectation_value = bra_notation @ op_mul_psi
+    print(f"Expectation Value: {expectation_value}")
+
+
+def tensor_expectation_value(sim: qcir_sim.TensorSimulation, gate: str, gate_idx: int):
+    psi = sim.tensor_state
+    psi /= np.linalg.norm(psi)
+    Op = sim._prepare_gate(gate, gate_idx)
+
+    op_mul_psi = np.einsum("ikj,ik->ij", Op, psi)
+    bra_notation = np.conjugate(np.transpose(psi))
+
+    expectation_value = np.einsum("ij,ji", bra_notation, op_mul_psi)
+    print(f"Expectation Value: {expectation_value}")
+
+
+if __name__ == "__main__":
+    args = parse_args()
+    simulation_type = args.sim_type
+    print(f"Using {simulation_type} simulation")
+
+    qubits = [qcir_sim.zero(), qcir_sim.one(), qcir_sim.one()]
+    if args.sim_type == "tensor":
+        sim = qcir_sim.TensorSimulation(qubits)
+    else:
+        sim = qcir_sim.VectorSimulation(qubits)
+
+    sim.add_gate("H", 1)
+    sim.add_gate("phase", 2)
+    sim.add_gate("H", 3)
+
+    if args.sim_type == "tensor":
+        tensor_expectation_value(sim, "H", 1)
+    else:
+        vector_expectation_value(sim, "H", 1)

examples/sampling/README.md

@@ -15,7 +15,7 @@ Run the program using:
 python examples/sampling/sampling.py -n 10 -t
 ```
 
-### Explaination
+### Explanation
 
 In a vector simulation of a n-qubit system, each element describes a possible combination of the state of qubit. For n-qubits, the state vector will have 2^n states.