Iterative Phase Optimization of Elementary Quantum Error Correcting Codes

Performing experiments on small-scale quantum computers is certainly a challenging endeavor. Many parameters need to be optimized to achieve high-fidelity operations. This can be done efficiently for operations acting on single qubits, as errors can be fully characterized. For multiqubit operations, though, this is no longer the case, as in the most general case, analyzing the effect of the operation on the system requires a full state tomography for which resources scale exponentially with the system size. Furthermore, in recent experiments, additional electronic levels beyond the two-level system encoding the qubit have been used to enhance the capabilities of quantum-information processors, which additionally increases the number of parameters that need to be controlled. For the optimization of the experimental system for a given task (e.g., a quantum algorithm), one has to find a satisfactory error model and also efficient observables to estimate the parameters of the model. In this manuscript, we demonstrate a method to optimize the encoding procedure for a small quantum error correction code in the presence of unknown but constant phase shifts. The method, which we implement here on a small-scale linear ion-trap quantum computer, is readily applicable to other AMO platforms for quantum-information processing.

Popular Summary

Achieving fully fledged quantum computers will require many iterative stages of increasing complexity. Initially, the goal of quantum computing was to achieve a high level of quantum control over one or two qubits. Nowadays, the target has become more challenging: construct small-sized quantum computers operating robustly and autonomously. However, environmental noise and imperfections affect the functioning of these prototype quantum computers and have thus far impeded scientists from realizing accurate, large-scale quantum computations. To overcome this obstacle, we have developed and successfully implemented quantum error correcting codes in small-scale quantum-information processors.

Achieving high accuracy in the initialization of quantum error correcting codes to store information requires precise control and the optimization of many parameters. Whereas quantum states and processes of single quantum bits can be characterized and optimized efficiently, this situation is no longer true for larger numbers of qubits. It is therefore highly desirable to develop efficient techniques to identify and characterize the dominant collective effects of noise in small- and medium-scale quantum processors. We introduce and experimentally demonstrate a new method to optimize the encoding procedure for small quantum error correction codes. Our method, which we implement in a linear ion-trap quantum computer involving seven qubits (trapped ${{}^{40}\mathrm{Ca}}^{+}$ ions) and evaluate numerically, makes it possible to detect and compensate in real time for unknown but constant phase shifts that deteriorate the fidelity of encoded quantum information.

Our technique is readily applicable to other physical platforms for quantum computing, including cold atoms and solid-state devices.

Figure 1

Schematics of the implemented seven-qubit quantum error correcting code and the encoding sequence. (a) One logical qubit is encoded in seven physical qubits forming a two-dimensional triangular planar structure of three plaquettes. The code space is defined as the simultaneous $+1$ eigenspace of a set of six four-qubit stabilizer operators associated with the plaquettes. (b) Physical qubits are encoded in (meta)stable electronic states of a string of seven ${{}^{40}\mathrm{Ca}}^{+}$ ions. The computational subspace of each physical qubit is spanned by the two electronic states $4^2{S}_{1/2}(m_j=-1/2)$ ($|1⟩$) and $3^2{D}_{5/2}(m_j=-1/2)$ ($|0⟩$). Another pair of states [$3^2{D}_{5/2}(m_j=-5/2)$ and $3^2{D}_{5/2}(m_j=-3/2)$] is used to spectroscopically decouple individual ion qubits. Red arrows indicate sequences of pulses that are applied to realize this coherent decoupling (see Ref. [16] for more details). Decoupled ions [indicated by dashed lines in (c)] ideally will not participate in subsequent dynamics, until they are recoupled, i.e., coherently transferred back into the computational subspace [solid lines in (c)]. This technique enables the application of entangling gate operations, which, in this setup, are implemented by illuminating the entire ion string by a global laser beam [18], to subsets of four qubits belonging to a given plaquette. (c) The logical qubit is encoded by coherently mapping the product input state $|1010101⟩$ onto the logical state ${|0⟩}_L$ [see Eq. (1)]. The quantum circuit combines spectroscopic decoupling and recoupling operations (white boxes) with plaquette-wise entangling operations that effectively create GHZ-type entanglement between qubits belonging to the same plaquette.

Figure 2

Experimental implementation of the phase optimization protocol. Here, the algorithm was applied to the intermediate state in the encoding sequence, which results from the application of the first two entangling operations acting on the qubits of the first (red) and second (blue) plaquettes of the planar seven-qubit quantum error correcting code. The resulting state (a), before the application of the iterative phase optimization technique, is characterized by positive values of $Z$-type plaquette stabilizer expectation values, which are maximal within the experimentally achieved accuracy of the encoding circuit [16]. On the other hand, $X$-type stabilizer expectation values have arbitrary values (positive on the first, negative on the second plaquette), indicating the presence of undesired, unknown relative phase shifts [see Eq. (6)]. In the first step of phase optimization (b), a $Z$ rotation of variable magnitude is applied to qubit 2, which results in a sinusoidal behavior of the expectation values of the stabilizers $⟨S_x^{(1)}⟩$ and $⟨S_x^{(2)}⟩$ [cf. Eqs. (8) and (9)], whereas the expectation value $⟨S_x^{(1)}{S}_x^{(2)}⟩$ not containing $X_2$ remains constant. For each scan, the stabilizer that takes part in the optimization procedure is highlighted by the bold line, and the corresponding maximum value is marked via the orange circle. After reading off and fixing ${\theta }_2$ to the value that maximizes $⟨S_x^{(1)}⟩$ (orange circle), a $Z$ rotation is applied to qubit 5 (c). This scan is used to fix ${\theta }_5$ to the value that maximizes $⟨S_x^{(2)}⟩$. Whereas, in principle, at this point one would proceed with the optimization of $⟨S_x^{(1)}{S}_x^{(2)}⟩$ by a $Z_1$-rotation scan, the data show that all three stabilizers, within experimental resolution, have already reached the maximum, indicating convergence of the protocol. This is also reflected by both $X$-type plaquette stabilizers now being positive and maximal (d), while the expectation values of $Z$-type stabilizers and of the logical $Z$ operator have remained unchanged over the application of the algorithm—compare (a) and (d). Experimental parameters: In each scan, different values for the phases characterizing the single-qubit rotations were applied with an elementary step size of $2\pi /10$. For each phase value, the experiment was repeated 200 times.

Figure 3

Experimental phase optimization of the complete seven-qubit quantum error correcting code. Here, the algorithm was applied to the final state resulting from the complete encoding sequence shown in Fig. 1, i.e., three entangling operations applied to the qubits belonging to the first (red), second (blue), and third (green) plaquette of the code. Initially, $X$-type stabilizer expectation values are nonmaximal (a), indicating the presence of unknown, relative phases in the desired target state. After two rounds of iteratively maximizing the seven expectation values of plaquette operators $⟨S_x^{(1)}⟩$, $⟨S_x^{(2)}⟩$, $⟨S_x^{(3)}⟩$, $⟨S_x^{(1)}{S}_x^{(2)}⟩$, $⟨S_x^{(2)}{S}_x^{(3)}⟩$, $⟨S_x^{(1)}{S}_x^{(3)}⟩$, and $⟨S_x^{(1)}{S}_x^{(2)}{S}_x^{(3)}⟩$, the algorithm converges to a set of compensation phases, $\mathit{\theta }=[{\theta }_1,\dots ,{\theta }_7]$, for which all $X$-type stabilizers assume maximal values. The individual phase value ${\theta }_i$ to the $Z$ rotation, which is adjusted to maximize the corresponding stabilizer expectation value under consideration (bold line), is indicated by the orange circle for each optimization step [see (b)–(i)]. Note that because of the periodicity in ${\theta }_i$, it is also possible to search for the minimum expectation value of the stabilizer under consideration and add the rotation angle $2\theta =\pi$; see (c) for an example. The $Z$- and $X$-type stabilizers of the logical state $|0{⟩}_L$ after two rounds of optimization steps are shown in (j). Intermediate steps of the second round of optimization are not shown. The experimental parameters are as specified in Fig. 2.

Figure 4

Number of iterations vs convergence threshold. This plot shows the scaling of the number of iterations required by PHOM with the tightness of the convergence criterion. As commented in the text, two figures of merit assess this, ${\delta }_1$ and ${\delta }_2$. The former is related to the distance between the sum of stabilizers and its maximum value, and the latter is associated with the maximum value among the distances for each stabilizer. The simulations have been done for the case of PHOM applied to individual mean values.

Figure 5

Larger instances of planar color codes. The 17-qubit code (a) encodes a logical qubit of logical distance $d=5$; the 31-qubit code (b) has distance $d=7$. Whereas the 17-qubit code would, at least in principle, allow for the correction of the undesired phases with physically quasilocal rotations, acting only on subsets of qubits belonging to the same plaquette, phase compensation for the 31-qubit case (and larger codes) would require nonlocal rotations involving qubits of several plaquettes.