Colloquium: Protecting quantum information against environmental noise

Quantum technologies represent a rapidly evolving field in which the specific properties of quantum mechanical systems are exploited to enhance the performance of various applications such as sensing, transmission, and processing of information. Such devices can be useful only if the quantum systems also interact with their environment. However, the interactions with the environment can degrade the specific quantum properties of these systems, such as coherence and entanglement. It is therefore essential that the interaction between a quantum system and the environment is controlled in such a way that the unwanted effects of the environment are suppressed while the necessary interactions are retained. This Colloquium gives an overview, aimed at newcomers to this field, of some of the challenges that need to be overcome to achieve this goal. A number of techniques have been developed for this purpose in different areas of physics including magnetic resonance, optics, and quantum information. They include the application of static or time-dependent fields to the quantum system, which are designed to average the effect of the environmental interactions to zero. Quantum error correction schemes were developed to detect and eliminate certain errors that occur during the storage and processing of quantum information. In many physical systems, it is useful to use specific quantum states that are intrinsically less susceptible to environmental noise for encoding the quantum information. The dominant contribution to the loss of information is pure dephasing, i.e., through the loss of coherence in quantum mechanical superposition states. Accordingly, most schemes for reducing loss of information focus on dephasing processes. This is also the focus of this Colloquium.

Figure 1

Initialization, processing, and detection (readout) of quantum information in the network model. The processing of information is performed by unitary transformations. Their sequence is determined by the algorithm.

Figure 2

Phase reversal and echo formation by an inversion ($\pi$) pulse applied to the qubit. The upper trace shows the pulses (rectangles) driving the evolution of the system as well as the average signal of an ensemble of spins as a function of time (solid green line). The middle part shows the orientation of two of the individual spin vectors at specific times, while the bottom trace shows their phase as a continuous function of time (dashed red vs solid blue lines).

Figure 3

Ground state sublevels of atomic cesium. The hyperfine structure as a function of magnetic field is shown, where $F$ and $m_F$ are the magnetic quantum numbers of the total spin operator and its $z$ component. The frequency of the clock transition ($m_F=0\leftrightarrow m_{F^{\prime }}=0$) is independent of the magnetic field (to first order).

Figure 4

The left-hand part shows the evolution of the phase due to a randomly fluctuating transition frequency, which corresponds effectively to a diffusion process. The right-hand part shows the decay of the coherence ${\rho }_{ij}$ of an ensemble of two-level systems suffering this random process.

Figure 5

Decays of the freely precessing magnetization and the Hahn echo of ${}^{13}\mathrm{C}$ nuclear spins in solid adamantane as a function of the evolution time. Both are proportional to the coherence ${\rho }_{12}$ of the spin density operator. The decay of the Hahn echo indicates that the environment that causes the dephasing is time dependent. The inset shows a Lorentzian-shaped environmental noise spectrum $S(\omega )$ (solid green line) together with the filter functions of the free (shaded gray curve) and the Hahn (transparent red shaded curve) evolution. In contrast to the free evolution, the Hahn filter vanishes at the origin ${|F(0,2\tau )|}^2=0$. From [8].

Figure 6

Interference of telegraph noise with the refocusing process. (a) The Hahn spin-echo sequence. (b), (c) The phase accumulation of the spins without and with a random single jump of the precession frequency of the spin, respectively. $f(t)$ gives the effective sign of the SE interaction during the sequence and ${\delta }_E(t)$ is the instantaneous coupling with the environment of two spins whose coupling to the environment is initially $+{\delta }_0$ (solid blue lines) and $-{\delta }_0$ (dashed red lines). The accumulated phase is shown to be fully refocused for the static case (b), but it is not refocused when a single jump occurs (c).

Figure 7

The sequence of $\pi$ rotations shown on top generates the echo train shown in the bottom trace. From [5].

Figure 8

Decay of the coherence of a single electron spin in the NV center of a diamond for different numbers of refocusing pulses. The different curves are displaced vertically to avoid overlap. Each data point represents the number of photons counted at the position of the last echo. As the number of pulses increases and the delay between the pulses decreases, the signal can be preserved for a longer time. From [169].

Figure 9

Dephasing time of ${}^{13}\mathrm{C}$ nuclear spins in adamantane as a function of the delay between the pulses for two different initial conditions (parallel and perpendicular to the rotation axis of the refocusing pulses). The number of refocusing pulses per unit of time increases then from right to left. The square symbols represent experimental data points and the curves are guides to the eye. From [8].

Figure 10

Fidelity of different pulse sequences after $N=20$ $\pi$ pulses as a function of the flip-angle error of the individual pulses. The different curves correspond to the CPMG, XY-4, and KDD sequences ([175]).

Figure 11

Fidelity of a DD sequence of $100\pi$ pulses as a function of two types of errors (offset, flip-angle). The resulting fidelity is color coded, with fidelities below 0.95 in white. For details see the text.

Figure 12

Quantum error correction scheme based on a three-qubit code. On input, the first (uppermost) physical qubit contains the logical information, while the ancilla qubits are initialized into the state $|0⟩$. The input information is then encoded into the logical qubit and the gate operation is performed. If a bit-flip error occurs, it can be corrected during the decoding process. At the end, the first physical qubit contains again the logical information.

Figure 13

Pulse sequence for a protected single-qubit gate. A single cycle of an XY-4 DD sequence is used to protect the gate operation. The labels $x$ and $y$ mark the rotation axes of the DD pulses and ${\mathcal{H}}_{Cn}$ the gate operations.

Figure 14

Effect of protecting a not gate by DD with a single XY-4 cycle (red squares) or an XY-8 cycle (green diamonds). The unprotected counterpart is represented by blue circles. The process fidelity of the gate operation is shown as a function of the gate duration. The results show that the dephasing due to the fluctuating environment is slowed down by several orders of magnitude by the combination of DD with the gate operation as shown in the inset. From [220].

Figure 15

Protected crot gate in a single NV center in diamond. The refocusing pulses are applied to the electron spin qubits as microwave pulses, while the nuclear spin qubit is rotated conditional on the electron spin qubits being initially in state $|1⟩$ by the radio-frequency pulses. The bottom part shows the measured population of the initial state with and without protection. In the absence of protection, it dephases rapidly, while the evolution in the presence of protection remains close to the ideal evolution. From [221].

Figure 16

Example of environmental noise spectra determined by DD noise spectroscopy. A narrow-band filter function scans the noise spectra (shaded curve). The reconstructed noise spectra are shown by the blue triangles and red circles for an unmodulated and a modulated environment, respectively. From [12].