Implementing quantum control for unknown subroutines

We present setups for the practical realization of adding control to unknown subroutines, supplementing the existing quantum optical scheme for black-box control with a counterpart for the quantum control of the ordering of sequences of operations. We also provide schemes to realize either task using trapped ions. These practical circumventions of recent no-go theorems are based on existing technologies. We argue that the possibility to add control to unknown operations in practice is a common feature of many physical systems. Based on the proposed implementations we discuss the apparent contradictions between theory and practice.

Figure 1

Circuit representation of no-go theorem. As shown in Refs. [1, 2], it is not possible to construct a quantum circuit that deterministically realizes $ctrl\text{−}U$ for an unknown unitary $U$, by using gates $A$ and $B$ that are independent of $U$, even if the ancilla can undergo arbitrary transformations $W_U$. From bottom to top the lines denote the system, the control qubit, and the ancilla. The mathematical meanings of the subcircuits in the dashed boxes, while different on both sides, are nonetheless independent of the remaining circuits.

Figure 2

Quantum control of a single device. In the scheme of Ref. [3], also discussed in Ref. [1], a single photon in the state ${\left|\varphi \right.〉}_c{\left|\psi \right.〉}_s$, where the polarization state ${\left|\varphi \right.〉}_c=\alpha {\left|H\right.〉}_c+\beta {\left|V\right.〉}_c$ serves as the control qubit and ${\left|\psi \right.〉}_s$ denotes the state of an additional degree of freedom, for instance, orbital angular momentum, is inserted into an interferometer. The polarizing beam splitters (PBSs) are configured to transmit and reflect horizontal and vertical polarization, respectively. The unknown unitary $U$ is assumed to act only on the system ${\left|\psi \right.〉}_s$. After recombination at the second PBS the initial state has been transformed to ${\left|\chi \right.〉}_{cs}=\alpha {\left|H\right.〉}_c{\left|\psi \right.〉}_s+\beta {\left|V\right.〉}_{c}U{\left|\psi \right.〉}_s$.

Figure 3

Controlling unknown unitaries with ions. The collective vibrational mode of two ions is used to create two submanifolds of electronic energy levels within each ion. The excitation of the vibration introduces a fixed energy shift between the qubit levels $\left|g\right.〉\left|1\right.〉$ and $\left|e\right.〉\left|1\right.〉$ on the right, with respect to $\left|g\right.〉\left|0\right.〉$ and $\left|e\right.〉\left|0\right.〉$ on the left. The auxiliary levels $\left|g^{\prime }\right.〉$ and $\left|e^{\prime }\right.〉$ (horizontal dashed lines) are chosen such that the hiding pulses $H_1$ and $H_2$, and the ${\sigma }_x$ pulses $S_g$ and $S_e$ have different frequencies and are off resonance with the qubit transitions between $\left|g\right.〉\left|n\right.〉$ and $\left|e\right.〉\left|n\right.〉$ $(n=1,2,...)$.

Figure 4

Circuit representation of no-go theorem. If only single uses of the unknown gates $U_f$ and $U_g$ are permitted, it is not possible to deterministically realize a generic circuit, i.e., with gates $A$, $B$, and $C$ that act on the system and are independent of $U_{f/g}$, which performs a controlled switch of the operations $U_f$ and $U_g$, that is, for which the order of the operations depends on the control qubit ${\left|\varphi \right.〉}_c$ (see Ref. [4]).

Figure 5

Quantum control of sequences. Our scheme enables the quantum control of the ordering in which single devices, labeled by $U_f$ and $U_g$, are applied to the system state ${\left|\psi \right.〉}_s$. As before, a single photon in the state ${\left|\varphi \right.〉}_c{\left|\psi \right.〉}_s$, where the polarization state ${\left|\varphi \right.〉}_c=\alpha {\left|H\right.〉}_c+\beta {\left|V\right.〉}_c$ serves as the control qubit and ${\left|\psi \right.〉}_s$ encodes an additional degree of freedom, such as orbital angular momentum, serves as the input. The PBSs are configured to transmit and reflect horizontal and vertical polarization, respectively. The half-wave plates $(\lambda /2)$ are configured to exchange vertical and horizontal polarization, while leaving the system state ${\left|\psi \right.〉}_s$ invariant. The unknown devices $U_f$ and $U_g$, on the other hand, are assumed to act on the system only. The beams are recombined at the rightmost PBS, where the final state is ${\left|\xi \right.〉}_{cs}=\alpha {\left|H\right.〉}_c{U}_g{U}_f{\left|\psi \right.〉}_s+\beta {\left|V\right.〉}_c{U}_f{U}_g{\left|\psi \right.〉}_s$.