Computational Advantage from the Quantum Superposition of Multiple Temporal Orders of Photonic Gates

Models for quantum computation with circuit connections subject to the quantum superposition principle have recently been proposed. In them, a control quantum system can coherently determine the order in which a target quantum system undergoes $N$ gate operations. This process, known as the quantum $N$-switch, is a resource for several information-processing tasks. In particular, it provides a computational advantage—over fixed-gate-order quantum circuits—for phase-estimation problems involving $N$ unknown unitary gates. However, the corresponding algorithm requires an experimentally unfeasible target-system dimension (super)exponential in $N$. Here, we introduce a promise problem for which the quantum $N$-switch gives an equivalent computational speedup with target-system dimension as small as 2 regardless of $N$. We use state-of-the-art multicore optical-fiber technology to experimentally demonstrate the quantum $N$-switch with $N=4$ gates acting on a photonic-polarization qubit. This is the first observation of a quantum superposition of more than $N=2$ temporal orders, demonstrating its usefulness for efficient phase estimation.

Popular Summary

Quantum circuits represent a breakthrough in computational power. This is achieved using superpositions of quantum states to speedup computations in classically impossible ways. Interestingly, models have been proposed in which the order in which the circuit gates are applied is itself in a quantum superposition. These are called quantum switches, and present advantages over standard quantum circuits, constituting a novel resource for quantum computation beyond the current quantum-circuit paradigm with fixed-gate orders. However, scaling up the quantum switch to many gates has been a challenge: all experimental implementations so far have attained quantum superpositions of only two gate orders, and theoretical proposals involving more gates, while valuable, have been unfeasible in practice.

In this paper, we achieve quantum superpositions of multiple gate orders. To accomplish this, we introduce a problem that is related to practical applications and whose best-known solution requires the quantum switch. Moreover, this new approach scales to many gates in a more realistic fashion. We experimentally demonstrate its feasibility by implementing a quantum switch with four gate orders using a photonic optical-fiber setup that solves instances of the problem.

This is the first experimental implementation of a superposition of more than two gate orders. This work brings quantum superposition of causal orders—originally a topic from abstract quantum foundations—to the realm of practical quantum computation. Crucial for our work to accomplish this is its built-in scalability features, which are key to make theoretical computational advantages concrete in practice.

Figure 1

(a) Abstract representation of the quantum $N$-switch for the case of $N=4$. The process, $W_4$ (light-gray region), can be thought of as an experimental setup (e.g., a quantum circuit or interferometer) through which the composite control-target system goes and with open slots for target-subsystem gates $U_i$ (dark-gray boxes) for $i=A$, $B$, $C$, or $D$ to be inserted. Inside $W_4$, the connections between these gates are coherently controlled by the control subsystem, an effect known as quantum control of gate orders. This property is a physical resource for certain quantum computations (phase-estimation problems), and $W_4$ is the resourceful object that bears it. The concatenation of $W_4$ with the inserted gates yields the quantum 4-switch gate $S_4$, a joint unitary operation on the composite system. (b) Concrete schematics of the specific variant of the quantum 4-switch process experimentally implemented in this work. The target subsystem undergoes the four-gate sequence in a quantum superposition (center) of $P=4$ different orderings (permutations of the string $ABCD$): $ABCD,BADC,CBDA,DACB$. Each permutation is shown individually in a different color and panel. (c) In the abovementioned computations, the target-subsystem gates are unknown. For the purpose of complexity analysis, they can be thought of as produced upon request by a quantum oracle $\mathcal{O}$. This takes as input $i=A$, $B$, $C$, or $D$ and outputs a black-box device implementing the unknown gate $U_i$. Each such call to the oracle counts as an oracle query. The $N$-switch process allows one to solve computational problems on the phase relationships between permutations of the black-box gates with considerably fewer oracle queries—i.e., lower query complexity—than any process with fixed (or classically controlled) gate connections.

Figure 2

(a) Illustration of our implementation of the quantum 4-switch gate ($S_4$). An input photon is divided coherently between four spatial modes using a four-core-fiber beam splitter (4CF-BS), placed between commercial multiplexer/demultiplexer (DMUX) units, as shown in (b). The four output modes are then sent to the quantum $4$-switch gate $S_4$. Each spatial mode is related to a unique permutation of the four unitary polarization operations applied by $S_4$ and indicated by a different color. The photons enter through the $\mathsf{I}\mathsf{N}$ side (right) and exit through the $\mathsf{O}\mathsf{U}\mathsf{T}$ side (left), where, for example, the notation “$\leftarrow A$” means “from A” and “$A\leftarrow$” means “to A.” One can follow a certain path by looking at the output labels. For instance, the green input mode enters in $C$ and continues to “$B$, then $D$, then $A$, and finally exits,” corresponding to the operation of the four polarization unitaries in the order $CBDA$. After $S_4$, the four spatial modes are then recombined using a second 4CF-BS. Each output 0–3 is connected directly to a single-photon detector (APD). The detection of a single-photon in the $y\mathrm{th}$ ($y=0,1,2,3$) output detector identifies in a single shot the phase relation $y$ of the four unitaries implemented in the quantum $4$-switch gate. See the main text and Sec. 7 for further details.

Figure 3

(a) A sequence of about 8 min of measurement results with our quantum $4$-switch process taken in real time. Measurements of 0.1 s duration are taken continuously, realized within the phase stabilization routine (see Sec. 7), in which the four sets of unitaries given each by the $y\mathrm{th}$ column of Table 1(a) are toggled randomly every minute. The number labels correspond to the columns of Table 1(a). Summary of experimentally obtained success probabilities to identify the commutation relations of the unitary operations in Table 1(a) [panel (b)] and Table 1(b) [panel (c)]. See the text for more details.

Figure 4

(a) Illustration of the 4CF launchers and the LCRs implementing the unitaries $U_i$. The LCRs are placed at the Fourier plane of the output coupling lenses. (b) Images recorded at the LCRs plane, of each core alone, as well as the output when all cores are connected, showing large spatial overlap between the cores modes. This guarantees that the $U_i$ are indistinguishable when applied in different orders.

Figure 5

Fixed-gate-order circuit that simulates the quantum $4$-switch process that is realized experimentally, i.e., with quantum control of the four gate sequences ${\mathrm{\Pi }}_0=U_D{U}_C{U}_B{U}_A$, ${\mathrm{\Pi }}_1=U_C{U}_D{U}_A{U}_B$, ${\mathrm{\Pi }}_2=U_A{U}_D{U}_B{U}_C$, and ${\mathrm{\Pi }}_3=U_B{U}_C{U}_A{U}_D$. Before and after each unitary $U_i$, a pair of controlled swap gates controls whether $U_i$ is applied to the target system or to an ancilla; the control qudit has dimension $P=4$, here represented as two qubits (with $x=0,1,2$, and $3$ encoded as $00,01,10$, and $11$, respectively). Filled circles indicate an operation conditioned on the ${\left|1\right.⟩}_c$ state; open circles indicate an operation conditioned on the ${\left|0\right.⟩}_c$ state. Conditioning on negation of certain states is also needed, as exemplified in the legend below the circuit.