Simulation of Hamming-weight-preserving circuits

This specialised backend in qibo supports the fast simulation of Hamming-weight-preserving circuits. Given a quantum state that only contains non-zero amplitudes associated with computational basis states of Hamming weight \(k\), the qibo.backends.hamming_weight.HammingWeightBackend works with a compressed representation of quantum states that is restricted to the subspace with fixed Hamming weight \(k\). The dimension of this subspace is \(d = \binom{n}{k}\), where \(n\) is the total number of qubits.

As for the other backends, the Hamming-weight backend can be set by specifying the engine used for calculation, i.e.

from qibo import set_backend

set_backend("hamming_weight", platform="numpy")

If not provided, the current global backend is used:

from qibo import set_backend
from qibo.backends import HammingWeightBackend

# setting numpy as the global backend
set_backend("numpy")
# the Hamming-weight backend will use the numpy backend as engine
backend = HammingWeightBackend()

A Hamming-weight-preserving circuit with Hamming weight \(k\) can be executed as the following example:

from qibo import Circuit, gates, set_backend
from qibo.backends import HammingWeightBackend

backend = HammingWeightBackend()

nqubits = 4
circuit = Circuit(nqubits)
circuit.add(gates.SWAP(0, 1))

result = backend.execute_circuit(circuit, weight=2)

# the above execution is equivalent to the following
# that uses the global backend and the ``weight`` kwarg
#
# set_backend("hamming_weight", platform="numpy")
# result = circuit(weight=2)

The results of execution of a circuit through this backend creates a qibo.quantum_info.hamming_weight.HammingWeightResult object that gives access to the final state through the qibo.quantum_info.hamming_weight.HammingWeightResult.state() method. The probabilities are computed by the qibo.quantum_info.hamming_weight.HammingWeightResult.probabilities() method. If there are no measurements in the circuit or the number of shots is \(0\), they are calculated from the statevector. Otherwise, they are calculated from the samples. The final measured samples and frequencies are accessible, respectively, through the qibo.quantum_info.hamming_weight.HammingWeightResult.samples() and the qibo.quantum_info.hamming_weight.HammingWeightResult.frequencies() methods.

It is also possible to recover the standard state representation with the qibo.quantum_info.hamming_weight.HammingWeightResult.full_state() method. Note, however, that this process is inefficient as it involves the construction of a statevector with size exponential in the number of qubits.