Experimental superposition of orders of quantum gates

Abstract

Quantum computers achieve a speed-up by placing quantum bits (qubits) in superpositions of different states. However, it has recently been appreciated that quantum mechanics also allows one to 'superimpose different operations'. Furthermore, it has been shown that using a qubit to coherently control the gate order allows one to accomplish a task--determining if two gates commute or anti-commute--with fewer gate uses than any known quantum algorithm. Here we experimentally demonstrate this advantage, in a photonic context, using a second qubit to control the order in which two gates are applied to a first qubit. We create the required superposition of gate orders by using additional degrees of freedom of the photons encoding our qubits. The new resource we exploit can be interpreted as a superposition of causal orders, and could allow quantum algorithms to be implemented with an efficiency unlikely to be achieved on a fixed-gate-order quantum computer.

Figure 1. Theoretical Concept.

(a) Given two unitary gates, U₁ and U₂, the circuit model allows us to wire them in one of two possible ways: either U₁ before U₂, or (b) U₂ before U₁. (c) Quantum mechanics allows us to coherently control both options, such that the qubit sees both U₁ before U₂, and U₂ before U₁. (d) The 2-SWITCH operation applies U₁ and U₂ to qubit 2 in both orders, as shown in panel c, dependent on the state of qubit 1. Unless at least one of U₁ and U₂ is used more than once, the 2-SWITCH operation cannot be implemented with standard circuit-model elements. To be explicit, the 2-SWITCH applies U₁U₂ to (the lower qubit) if the upper qubit is in , and U₂U₁ to if the upper qubit is in . Measuring the state of qubit 1 in the basis allows one to unambiguously decide if U₁ and U₂ commute or anti-commute with only a single use of each gate. In this circuit, H is the Hadamard gate, and .

Figure 2. Experimental Implementation.

Our optical implementation to distinguish whether a pair of unitary gates commute or anti-commute with only a single copy of each gate. The photons for our experiment are generated in a separable polarization state using a Sagnac source (a). One photon is used as a herald, and the second is fed into the interferometer (b). The unitary gates in question are each implemented with three waveplates, and act on the polarization of single photons.

Figure 3. Results for Pauli Gates.

Experimental data showing the probability with which the photon exits from a port when determining if a pair of random gates commute or anti-commute. The blue bars are the experimentally observed probabilities for the photon to exit port 1, and the green bars to exit port 0. If the gates commute, then, ideally, the photon should always exit port 0, while if they anti-commute the photon should exit port 1. The x axis is labelled with the choice of U₁ and U₂, where , is the identity, X=σ_x, Y=σ_y, and Z=σ_z. The average success rate (probability to exit the ‘correct port') of these data is 0.973±0.016.

Figure 4. Results for Random Gates.

Experimental data showing the probability with which the photon exits from a port when determining if a pair of random gates commute or anti-commute. 50 commuting and 50 anti-commuting pairs of gates were tested, of which 10 for each case are shown here. The full data set is presented in the Supplementary Fig. 2. The data representation in this figure follows the same convention as in Fig. 3. However, here the x axis is labelled Ai for anti-commuting case number i, and Ci for commuting case number i.