Quantum circuits

The module Circuits provides a way to construct and simulate circuits. Initialise an empty circuit with Circuit(n) where n is the number of qubits. Then, add gates to the circuit with push!(circuit, gate, qubits...) where gate is a string representing the gate and qubits are the qubits the gate acts on. For example, a CNOT gate with control qubit 1 and target qubit 2 is added to the circuit c with push!(c, "CNOT", 1, 2). Depolarizing noise can be added to the circuit with push!(c, "Noise"). The noise amplitude can be set with c.noise_amplitude=p.

Lets construct a CCX gate (Toffoli gate) out of CNOT and single qubit gates.

using PauliStrings
using PauliStrings.Circuits

function noisy_toffoli()
    c = Circuit(3)
    push!(c, "H", 3)
    push!(c, "CNOT", 2, 3); push!(c, "Noise")
    push!(c, "Tdg", 3)
    push!(c, "CNOT", 1, 3); push!(c, "Noise")
    push!(c, "T", 3)
    push!(c, "CNOT", 2, 3); push!(c, "Noise")
    push!(c, "Tdg", 3)
    push!(c, "CNOT", 1, 3); push!(c, "Noise")
    push!(c, "T", 2)
    push!(c, "T", 3)
    push!(c, "CNOT", 1, 2); push!(c, "Noise")
    push!(c, "H", 3)
    push!(c, "T", 1)
    push!(c, "Tdg", 2)
    push!(c, "CNOT", 1, 2); push!(c, "Noise")
    return c
end

noisy_toffoli (generic function with 1 method)

We can plot the circuit using QuantumCircuitDraw.jl:

using QuantumCircuitDraw
c = noisy_toffoli()
paulistrings_plot(c)

The circuit can be compiled to a unitary matrix with compile(c). Before compiling, set c.noise_amplitude to the desired noise amplitude and c.max_strings to the number of strings to keep at each step. Compile will multiply the gates in the circuit from left to right, apply the noise using add_noise and trim the operator at each step using trim.

Lets check our Toffoli gate against the built-in CCX gate in Circuits:

using PauliStrings.Circuits
c = noisy_toffoli()
c.noise_amplitude = 0
U1 = compile(c)
U2 = CCXGate(3,1,2,3)
println(opnorm(U1-U2))

1.8350455610866623e-15

We can also compute expectation values of states in the computational basis with expect(c, state_out) and expect(c, state_in, state_out) . Let's compute the expectation values of the output states of the Toffoli gate when the input state is $|111\rangle$:

in_state = "111"
out_states = ["000", "001", "010", "011", "100", "101", "110", "111"]
p = [expect(c, in_state, out_state) for out_state in out_states]
using Plots
bar(out_states, p, legend=false, xlabel="out state", ylabel="<out|U|in>")

Same as above but seting the noise amplitude to 0.05:

c.noise_amplitude = 0.05
p = [expect(c, in_state, out_state) for out_state in out_states]
bar(out_states, p, legend=false, xlabel="out state", ylabel="<out|U|in>")