Gate teleportation

“Entanglement-assisted communication becomes entanglement-assisted computation” - Furusawa [7]

In the quantum state teleportation algorithm, the quantum state is transferred from the sender to the receiver exactly. However, quantum teleportation can be used in a much more powerful manner, by simultaneously processing and manipulating the teleported state; this is known as gate teleportation.

But the biggest departure from its namesake is the method in which the gate to be ‘teleported’ is applied; rather than applying a quantum unitary directly to the first qumode in the system, the unitary is applied via the projective measurement of the first qumode onto a particular basis. This measurement-based approach provides significant advantages over applying unitary gates directly, for example by reducing resources, and in the application of experimentally hard-to-implement gates [7]. In fact, gate teleportation forms a universal quantum computing primitive, and is a precursor to cluster state models of quantum computation [8][9].

CV implementation

First described by Gottesman and Chuang [8] in the case of qubits, gate teleportation was generalized for the CV case by Bartlett and Munro in 2003 [10]. In an analogous process to the discrete-variable case, you begin with the algorithm for local state teleportation:

../_images/gate_teleport1.svg


Note that:

  • Unlike the spatially-separated quantum state teleportation we considered in the previous section, local teleportation can transport the state using only two qumodes; the state we are teleporting is entangled directly with the squeezed vacuum state in the momentum space through the use of a controlled-phase gate.
  • The state is then teleported to qumode \(q_1\) via a homodyne measurement in the computational basis (the position quadrature).
  • Like in the previous section, to recover the teleported state exactly, we must perform Weyl-Heisenberg corrections to \(q_1\); here, that would be \(F^\dagger X(m)^\dagger\). However, for convenience and simplicity, we write the circuit without the corrections applied explicitly.

Rather than simply teleporting the state as-is, we can introduce an arbitrary unitary \(U\) that acts upon \(\ket{\psi}\), as follows:

../_images/gate_teleport2.svg


Now, the action of the unitary \(U\) is similarly teleported along with the initial state — this is a trivial extension of the local teleportation circuit. In order to view this in as a measurement-based universal quantum computing primitive, we make a couple of important changes:

  • The inverse Fourier gate is absorbed into the measurement, making it a homodyne detector in the momentum quadrature
  • The unitary gate \(U\), if diagonal in the computational basis (i.e., it is of the form \(U=e^{i f(\hat{x}^i)}\)), commutes with the controlled-phase gate (\(CZ(s)=e^{i s ~\hat{x_1}\otimes\hat{x_2}/\hbar}\)), and can be moved to the right of it. It is then also absorbed into the projective measurement.
../_images/gate_teleport3.svg


Additional gates can now be added simply by introducing additional qumodes with the appropriate projective measurements, all ‘stacked vertically’ (i.e., coupled to the each consecutive qumode via a controlled-phase gate). From this primitive, the model of cluster state quantum computation can be derived [9].

Note

What happens if the unitary is not diagonal in the computational basis? In this case, feedforward is required; additional qumodes and projective measurements are introduced, with successive measurements dependent on the previous result [11].

Blackbird code

Consider the following gate teleportation circuit,

../_images/gate_teleport_ex.svg


Here, the state \(\ket{\psi}\), a squeezed state with \(r=0.1\), is teleported to the final qumode, with the quadratic phase gate (Pgate) \(P(s)=e^{is\hat{x}^2/2\hbar}\) teleported to act on it - with the quadratic phase gate chosen as it is diagonal in the \(\x\) quadrature. This can be easily implemented using the Blackbird quantum circuit language:

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# create initial states
Squeezed(0.1) | q[0]
Squeezed(-2)  | q[1]
Squeezed(-2)  | q[2]

# apply the gate to be teleported
Pgate(0.5) | q[1]

# conditional phase entanglement
CZgate(1) | (q[0], q[1])
CZgate(1) | (q[1], q[2])

# projective measurement onto
# the position quadrature
Fourier.H | q[0]
MeasureX | q[0]
Fourier.H | q[1]
MeasureX | q[1]

Some important notes:

  • As with the state teleportation circuit above, perfectly squeezed vacuum states are not physically realizable; preparing the states with a squeezing factor of \(|r|=2\) (\(\sim 18\text{dB}\)) is a reasonable approximation.
  • The Blackbird notation Operator.H denotes the Hermitian conjugate of the corresponding operator.
  • Here, we do not make the corrections to the final state; this is left as an exercise to the reader. For additional details, see the gate teleportation commutation relations derived by van Loock [11].

To easily check that the output of the circuit is as expected, we can make sure that it agrees with the (uncorrected) state

\[X({q_1})FP(0.5)X(q_0)F \ket{z}\]

which can be generated as follows:

Squeezed(0.1) | q[3]
Fourier       | q[3]
Xgate(q[0])   | q[3]
Pgate(0.5)    | q[3]
Fourier       | q[3]
Xgate(q[1])   | q[3]

Note

A fully functional Strawberry Fields simulation containing the above Blackbird code is included at examples/gate_teleportation.py.