Quantum error correction using weak measurements

For a binary weak measurement,

where $\xi(t)$ is a white noise with $\langle\xi(t)\rangle=0$ and $\langle\xi(t)\xi(t^{\prime})\rangle=\delta(t-t^{\prime})$. During weak measurement, $w_{0}-w_{1}$ can be experimentally observed along any quantum trajectory.

In recent years, weak measurements have been implemented experimentally for superconducting transmon qubits Vijay et al. (2012); Murch et al. (2013). A transmon qubit is essentially a tunable nonlinear quantum oscillator made of two Josephson junctions in a superconducting loop shunted by a capacitor. The lowest two energy levels of the nonlinear oscillator are used as a qubit. The qubit is kept in a microwave cavity with dispersive coupling, and its weak measurements are carried out by probing the cavity by a microwave signal. For weak measurement of a transmon, the signal observed by the apparatus is a current $I_{m}$, and $w_{0}-w_{1}$ is obtained by scaling it suitably. We adopt the convention that the ideal measurement current is $\pm\frac{\Delta I}{2}$ for the measurement eigenstates $P_{0}$ and $P_{1}$. Then the scaled measurement current, $\frac{2I_{m}}{\Delta I}=w_{0}-w_{1}$, provides an estimate of $\textrm{Tr}(\rho P_{0})-\textrm{Tr}(\rho P_{1})$.

The redundancy of the encoding QEC protocol allows extraction of the syndrome information from the encoded state without disturbing the encoded information. The simplest code to correct a single bit-flip error encodes the logical qubit into three physical qubits. Parities of qubit-pairs provide the syndrome information, which is extracted into two additional ancilla qubits, using $C$-NOT logic gates as illustrated in Fig.1. These parities are then used to apply controlled inverse logic operations to eliminate the error. (For the sake of simplicity, we have assumed that all gate operations are perfect, and the qubits are exposed to the error perturbations only before the measurement and feedback steps. A more elaborate fault-tolerant prescription can take care of errors occurring anywhere during the evolution.) After using the parities for the feedback, the ancilla qubits need to be disentangled from the encoded state, and that is achieved by resetting the ancilla qubits. This resetting is an irreversible step, and it reduces the fidelity of the encoded quantum state. Different methods have been proposed for resetting the ancilla qubits Reed et al. (2010); Geerlings et al. (2013), and they can be applied equally well after strong or weak measurements.

Initially the quantum system is in the logical subspace. Under the influence of undesired disturbances, it moves out of the logical subspace along some of the error directions. The syndrome operators are designed to reveal the information about the error without disturbing the encoded signal. The binary syndrome operators have a $+1$ eigenvalue for the logical subspace and a $-1$ eigenvalue for some of the error states. The complete error information is constructed from a collection of binary syndrome measurements.

The measurement evolution, described by Eqs. (1,2,3), can be expressed in the Itô form as Gisin (1984):

Here the stochastic Wiener increment obeys $\langle dW(t)\rangle=0$ and $\langle(dW(t))^{2}\rangle=dt$. For any initial state $z_{0}$, the time evolution of the quantum trajectory distribution is known Gisin (1984). After time $\tau$, the initial $\delta$-function distribution becomes a sum of two Gaussians with areas $p_{0}$ and $1-p_{0}$, centered at $z_{0}+g\tau$ and $z_{0}-g\tau$ respectively, and with a common variance $g\tau$.

The unperturbed logical space state has $p=1$ or $z=\infty$. Due to errors, the state gets shifted to some finite $z_{0}$, and the feedback task is to move it back to $z=\infty$. The syndrome measurement is designed to estimate $z_{0}$, but ends up shifting the state more in the process. To figure out what feedback to apply, with only incomplete information available, we make the following approximations:
(i) We approximate the measurement evolution as a binary walk on a line, producing $\delta$-function distributions at $z_{0}\pm g\tau$ with probabilities $p_{0}$ and $1-p_{0}$. For each step of a quantum trajectory, only one of the two possibilities occur, i.e. the post-measurement state is either $z_{0}+g\tau$ or $z_{0}-g\tau$. Of course, $g\tau$ is known from the properties of the measurement apparatus.
(ii) The current measured by the apparatus provides an average value of the signal over the measurement duration; so we use $\frac{2I_{m}}{\Delta I}$ to estimate either $z_{0}+\frac{g\tau}{2}$ or $z_{0}-\frac{g\tau}{2}$.
(iii) The measured current $I_{m}$ is a combination of $\textrm{Tr}(\rho\sigma_{z})$ and white noise. We next assume that if $I_{m}>0$ then the post-measurement state is $z_{0}+g\tau$, and if $I_{m}<0$ then the post-measurement state is $z_{0}-g\tau$.

With these assumptions, we convert $I_{m}$ into a rotation angle on the Bloch sphere, and design the feedback transformation. When $I_{m}>0$, the act of measurement moves the state from $z_{0}$ to $z_{0}+g\tau$, which is closer to the logical subspace corresponding to $z=\infty$. We then opt to apply no additional feedback operation. When $I_{m}<0$, we construct the feedback operation as follows. In terms of polar coordinates on the Bloch sphere,

We estimate the post-measurement state location as,

Here the replacement of $\tanh(z_{0}-\frac{g\tau}{2})$ by $\frac{2I_{m}}{\Delta I}$ is the most severe approximation, due to which the last line of the previous equation is not restricted to the interval $[-1,1]$. Whenever the value goes outside this interval, we replace it by $-1,+1$ corresponding to the value being less than $-1$ or greater than $+1$ respectively.

The rotation angle $\theta$ represents the combined shift of the quantum state due to error and measurement. This shift can be reversed by applying an inverse rotation by $-\theta$. The binary measurement does not determine the polar angle $\phi$ of the shift on the Bloch sphere. To estimate $\phi$, knowledge of off-diagonal elements of $\rho$ is needed, which is not available to us. (In case of projective measurements, $\theta=0,\pi$ and ignorance of $\phi$ does not matter.) Knowledge of $\phi$ is necessary to decide around which axis in the X-Y plane the feedback rotation $-\theta$ should be applied. Without that knowledge, we ad hoc choose the X-axis as the rotation axis. Consequently, the feedback may correct the error or may make it worse. Whenever the error gets worse, it would become easier to detect, and then to correct, in the next iteration. We hope that the procedure converges with repeated error correction steps with high fidelity, and our simulation results support that.

In weak measurements, the measured current, given by Eq.(3), is a noisy current. We can reduce the noise, and hence increase the accuracy of Eq.(7), by accumulating the signal over as long evolution time as possible:

Use of the integrated current $\tilde{I}_{m}$ as the signal, instead of the original current $I_{m}$, is tantamount to using a stronger interaction strength $g$. Also, a non-uniform integration weight in the definition of $\tilde{I}_{m}$ would lose some information, and so is not worthwhile. Once the feedback has been applied, then we have to erase all the current history, and wait until ample new current data is accumulated, before deciding on the next feedback operation. All this is equivalent to saying that we apply feedback only when we have sufficient information, and don’t disturb the system otherwise.

Now consider the situation where the syndrome consists of two commuting binary measurements, and $I_{m}^{1}$ and $I_{m}^{2}$ are the corresponding measured currents. The physical Hilbert space can be divided into four sectors corresponding to the measurement projectors $P_{0}^{1}P_{0}^{2}$, $P_{1}^{1}P_{0}^{2}$, $P_{0}^{1}P_{1}^{2}$, $P_{1}^{1}P_{1}^{2}$, where the superscript denotes the measurement number. For each binary measurement, we determine the rotation angle as in the previous subsection:

where $[\ldots]_{r}$ denotes reduction of the value to the interval $[-1,1]$, as described after Eq.(7). The feedback operation is then constructed from these angles. The syndrome definition tells us the location of the error in the multi-qubit register; so we determine the location of the error from the signs of $\{I_{m}^{k}\}$. For $\sigma_{z}$-measurements, we do not have knowledge of the polar angles $\phi_{k}$, and as before, we ad hoc choose the $X$-axis as the rotation axis.

When all the $I_{m}^{k}$ are positive, we do not apply any feedback operation. When only one of the $\{I_{m}^{k}\}$ is negative, we use the inverse rotation $-\theta_{k}$ as the feedback operation. When more than one $\{I_{m}^{k}\}$ are negative, assuming each $I_{m}^{k}<0$ to be an equal diagnostic of the error, we estimate movement of the state out of the logical subspace by averaging the corresponding $\cos\theta_{k}$. The feedback operation is then the inverse rotation $-\bar{\theta}$ obtained from the averaged projection. For instance, when $I_{m}^{1}<0$ and $I_{m}^{2}<0$,

This is an empirical prescription, but numerically we find it to be a good approximation.

Consider a single qubit system initialized in the state $|0\rangle$ that undergoes a bit-flip error. The error Hamiltonian is

where $\sigma_{x}$ is the Pauli operator representing the bit-flip error, and the error coupling $\gamma$ is chosen as a Gaussian random number with zero mean and variance $\alpha^{2}$. (We choose this form of $\gamma$ as a likely scenario in actual experiments.) Evolution under the error Hamiltonian for time $\tau$ changes the state of the system as

where $U=e^{-i\gamma\sigma_{x}\tau}$. For a system initialized in the state $|0\rangle$, Eq.(17) simplifies to

Averaging over the distribution of the error coupling $\gamma$, the averaged density matrix becomes:

Fidelity of the quantum state evolves as

and eventually the system reaches the completely mixed state.

Choosing $\gamma$ as a Gaussian random variable is an assumption, and the error distribution may be different in a different experimental setting. As another example, we consider the binary error distribution case, again with zero mean and variance $\alpha^{2}$. Then the error coupling is either $+\alpha$ or $-\alpha$ with equal probability. In this case, the averaged density matrix after time evolution $\tau$ becomes:

and the fidelity is

Although we use the Gaussian error distribution in our simulations throughout this work, we note that the choice between the Gaussian or the binary error distribution does not matter much for small values of $\alpha\tau$, both giving essentially the same fidelity as shown in Fig. 2.

To protect a single qubit system against bit-flip error, we redundantly encode it in a three-qubit register as:

The states $|0\rangle_{L}$ and $|1\rangle_{L}$ are basis vectors of the logical space, while $|000\rangle_{P}$ and $|111\rangle_{P}$ are basis vectors of the physical space. A general logical state to be protected is

with $|a|^{2}+|b|^{2}=1$. In our simulations, we have chosen the initial state as $a=1$ to simplify calculations; the linearity of evolution ensures that when the protocol works for the basis states, it will also work for any superposition of basis states. Our measured fidelity of the encoded state is thus

We diagnose the bit-flip errors using the syndrome operators $ZZI$ (i.e. $\sigma_{z}\otimes\sigma_{z}\otimes I$) and $IZZ$ (i.e. $I\otimes\sigma_{z}\otimes\sigma_{z}$).

In the three-qubit Hilbert space, the independent bit-flip error Hamiltonian is:

where the error couplings $\gamma_{i}$ are independent Gaussian random numbers with zero mean and variance $\alpha^{2}$.

Let the measurement currents for $ZZI$ and $IZZ$ be $I_{m}^{1}$ and $I_{m}^{2}$ respectively. We take the feedback Hamiltonian to be

where $\lambda_{i}$ are the feedback couplings. We estimate the feedback rotation angles as described in Section II.C. The feedback couplings are then:

In case of projective measurement (i.e. $g\tau\rightarrow\infty$), $\cos\theta_{i}=\pm 1$, and the fidelity of the quantum state after applying the feedback can be analytically determined as:

From Eqs. (20) and (29), we find that

and therefore projective measurement error correction always improves fidelity, irrespective of the value of $\alpha\tau$.

In our simulations, we varied $\alpha\tau$ (by holding $\alpha$ fixed and varying $\tau$), and observed how fidelity changes as the measurement strength $g\tau$ is varied. Our results, averaged over $10^{5}$ trajectories to cut down the statistical errors, are shown in Fig. 2. We observe that for large $g\tau$, the fidelity approaches the result obtained by performing projective measurement error correction. For small $g\tau$, the measurement current $I_{m}$ fluctuates heavily, leading to uncertain feedback that spoils the fidelity. Increasing $g\tau$ reduces these fluctuations, and the measurement accumulates sufficient information to improve fidelity. As a consequence, it is not desirable to perform error correction when $g\tau$ is rather small.

It can also be observed from Fig. 2 that fidelity improvement varies, depending on the relative strengths of $g$ and $\alpha$. To illustrate that, we have plotted respectively in Figs. 3, 4, the upper and lower bounds on $\alpha\tau$ for different values of $g\tau$, between which $1-f^{(3)}$ reduces by at least a factor of two after applying error correction. These two bounds arise for different physical reasons, and can be understood as follows.

When the measurement coupling decreases, we need to evolve the system for long $\tau$ to accumulate sufficient information. Within the same duration, to keep the overall error under control, we need to decrease $\alpha$. As a result, the upper bound on $\alpha\tau$ decreases with decreasing $g\tau$, as displayed in Fig. 3 (this effect is implicit in Fig. 2). For projective measurement error correction, the upper bound on $\alpha\tau$, calculated analytically using Eqs.(20) and (29), is 0.4905. Fig. 3 shows that $\alpha\tau$ approaches this value as we increase $g\tau$. Beyond this threshold, multiple errors overwhelm the error correction process and prevent improvement in fidelity.

The lower bound on $\alpha\tau$ results from the error correction protocol becoming noisy, when inadequate information is extracted by the weak measurement. In case of projective measurement, the post-measurement state is known precisely, which allows perfect error correction. But for smaller $g\tau$, it is difficult to accurately estimate the qubit state from the measurement current $I_{m}$ (the approximations described in Section II.B become far from precise), unless it is sufficiently perturbed by a large $\alpha\tau$. Consequently the lower bound on $\alpha\tau$ increases with decreasing $g\tau$, as displayed in Fig. 4 (this effect is also visible in Fig. 2).

From our results, we find that the two bounds on $\alpha\tau$ meet around $g\tau=5.25$. Attempts to correct errors with smaller values of $g\tau$ are pointless; the best option in such situations is not to perform any error correction.

In the previous subsection, we described how to correct single bit-flip errors. But a complete quantum error correction protocol has to correct all errors that may occur. For a single qubit, the complete error Hamiltonian can be written as:

Here $\sigma_{x}$ represents a bit-flip error, $\sigma_{z}$ represents a phase-flip error, and $\sigma_{y}$ represents both of them occurring together. We take the error couplings $\gamma_{i}$ to be independent Gaussian random numbers with zero mean and variance $\alpha^{2}$. Assuming that the system is initialized in the state $|0\rangle$, the density matrix after evolution with the error Hamiltonian $H_{E}$ for time $\tau$ becomes,

when averaged over the distributions of $\gamma_{i}$. The fidelity of the quantum state therefore evolves as

If the error distribution is taken to be a binary distribution, instead of a Gaussian distribution, the averaged density matrix becomes

which has the fidelity

For small values of $\alpha\tau$, Eqs.(32) and (34) give almost the same fidelity as depicted in Fig. 5, and the choice of error distribution doesn’t matter much.

To protect the single logical qubit from arbitrary error, we encode it in a five-qubit physical register Laflamme et al. (1996); Bennett et al. (1996), according to Preskill :

where $\bar{X}=XXXXX$. In our simulations, without loss of generality, we choose the initial state to be $|00000\rangle$, which makes our measured fidelity of the encoded state

We diagnose all single Pauli errors using the four syndrome operators:

For the five-qubit register, the independent error Hamiltonian can be written as:

where the error couplings $\gamma_{i}$ are independent Gaussian random numbers with zero mean and variance $\alpha^{2}$.

Our feedback Hamiltonian is:

where $\lambda_{i}$ are the feedback couplings. With four different binary syndrome measurements, there are 16 possible outcomes. The one with all four currents positive stands for “no error”, while the other 15 possibilities correspond to single qubit $X,Y,Z$ errors. Different combinations of the measured currents determine the corresponding non-zero $\lambda_{i}$, as listed in the following table.

We estimate the feedback rotation angle by extending the procedure described in Section II.C to four binary measurements. At every evolution step, only one (or none) of the fifteen feedback couplings is non-zero, determined by its unique syndrome signature. The non-zero rotation angle always equals $-\frac{\bar{\theta}}{2\tau}$.

In our simulations, we once again varied $\alpha\tau$ (by holding $\alpha$ fixed and varying $\tau$), and observed how the fidelity changes as the measurement strength $g\tau$ varies. Our results, averaged over $10^{5}$ trajectories to control the statistical errors, are shown in Fig. 5. We notice the same overall features as in the case of the bit-flip error. For large $g\tau$, as expected, the fidelity approaches the value for projective measurement error correction. For small $g\tau$, large fluctuations of the measurement current $I_{m}$ make the feedback uncertain and spoil the fidelity. With increasing $g\tau$, these fluctuations reduce and the fidelity improves, while error correction with rather small $g\tau$ is to be avoided.

Improvement of the fidelity varies depending on the relative strength of $g$ and $\alpha$, which can be observed in Fig.5. To make that more explicit, we have plotted respectively in Figs. 6 and 7, the upper and lower bounds on $\alpha\tau$ for different values of $g\tau$, between which $1-f^{(5)}$ reduces by at least a factor of two after applying error correction. The physical reasons for these bounds are the same as those described for the bit-flip error correction scheme in the previous subsection. The upper bound specifies the threshold beyond which error correction fails due to multiple errors, and the lower bound signifies the minimum information to be extracted by measurement in order to perform error correction. The two bounds meet around $g\tau=9.1$, and it is not worthwhile to attempt error correction for smaller values of $g\tau$.

In case of projective measurement, we numerically find that error correction cannot improve fidelity beyond $\alpha\tau=1.375$. When we demand a factor of two improvement in $1-f^{(5)}$, this threshold decreases to $\alpha\tau=1.025$. Our results approach this upper bound rather slowly as $g\tau$ increases. This behaviour sharply contrasts with the fast approach to the upper bound in case of the bit-flip error correction protocol. It may be that the considerably larger error subspace of the five-qubit register requires the measurement to extract more information in order to cut down the error.