Characterizing Quantum Instruments: From Nondemolition Measurements to Quantum Error Correction

In advanced quantum processors, quantum operations are increasingly processed along multiple in-sequence measurements that result in classical data and affect the rest of the computation. Because of the information gain of classical measurements, nonunitary dynamical processes can affect the system, which common quantum channel descriptions fail to describe faithfully. Quantum measurements are correctly treated by so-called quantum instruments, capturing both classical outputs and postmeasurement quantum states. Here we present a general recipe for characterizing quantum instruments and demonstrate its experimental implementation and analysis. Thereby the full dynamics of a quantum instrument can be captured, exhibiting details of the quantum dynamics that would be overlooked with standard techniques. For illustration, we apply our characterization technique to a quantum instrument used for the detection of qubit loss and leakage, which was recently implemented as a building block in a quantum error-correction (QEC) experiment [Nature 585, 207 (2020)]. Our analysis reveals unexpected and in-depth information about the failure modes of the implementation of the quantum instrument. We then numerically study the implications of these experimental failure modes on QEC performance, when the instrument is employed as a building block in QEC protocols on a logical qubit. Our results highlight the importance of careful characterization and modeling of failure modes in quantum instruments, as compared to simplistic hardware-agnostic phenomenological noise models, which fail to predict the undesired behavior of faulty quantum instruments. The presented methods and results are directly applicable to generic quantum instruments and will be beneficial to many complex and high-precision applications.

Popular Summary

Experimental advances in the field of quantum computation progress rapidly, addressing ever more complex applications. Those tasks increasingly include situations where we interrupt the computation with a measurement and then use that information for the rest of the computation. Owing to its destructive nature, the outcome of a quantum measurement results in classical data. Operations including in-sequence measurements therefore deviate from simple linear evolution. Such applications prominently feature in quantum networks, questions of causality, and quantum error correction.

Advanced operations can therefore not be described by existing methods. However, a failure to take into account the full dynamics of these operations can lead to the accumulation of errors that is fatal in high-performance applications. We show how to correctly treat such quantum classical operations by means of so-called quantum instruments capturing both classical and quantum degrees of freedom. We deliver a complete toolbox on the tomographic characterization of generic instruments.

As a working example, we experimentally illustrate a quantum instrument aimed at detecting lost qubits, which is a crucial task for fault-tolerant quantum computers. Our analysis sheds light on surprising nonidentity dynamics of the loss detection unit, whose influence on quantum error correction we further quantify in numerical simulations. The results underline the importance of observing and properly characterizing these dynamics, which, if ignored, drastically deteriorate the information in logical qubits.

The results thereby offer a new paradigm toward characterization of advanced quantum operations. Our methods are widely applicable to other quantum instruments and a variety of physical platforms and thereby help to reveal detailed information about dynamics that remains uncovered using conventional methods.

Figure 1

Example of a quantum instrument. The operation $U$ realizes a generic quantum instrument on the system in initial state ${\rho }_q$ and writes index $j$ of the applied operation into the state of an ancilla initialized in state ${|0⟩}_a$. The ancilla is finally projected onto the computational basis to read out the classical index.

Figure 2

Ion-trap quantum processor and qubit loss detection unit as the example quantum instrument. (a) Schematic of our ion-trap quantum processor, where each ion resides along a linear string representing a single qubit. Quantum gate operations are realized upon coherent laser-ion interaction using tightly focused beams addressing single ions for local gates (bright red) and a pair of ions for entangling gates (dark red). Readout is performed via collective fluorescence detection (DET). See the text for details. (b) A QND qubit loss detection unit as our application example for a quantum instrument. The system qubit is encoded in the computational subspace $\{{|0⟩}_q,{|1⟩}_q\}$ and is affected by loss to a third level ${|2⟩}_q$. For details, see the text. (c) A quantum erasure channel implemented by first inducing partial loss from ${|0⟩}_q$ followed by its detection using the gadget from (b). Conditional on the qubit not being lost, the same partial loss is induced from ${|1⟩}_q$ and subsequently detected.

Figure 3

Comparison of trace-constrained and trace-unconstrained tomography for the nonunitary map ${\mathcal{E}}_0$ from Fig. 2. We compute the total variation distance between directly measured frequencies and those predicted from the reconstructed Choi operators. Standard MLE from Eq. (5), referred to as the trace-constrained approach, increasingly fails to capture the underlying dynamics for higher loss probabilities, whereas the trace-unconstrained approach from Eq. (4) matches the predicted outcomes. Error bars correspond to one standard deviation of statistical uncertainty due to quantum projection noise. Further notes characterizing erroneous effects owing to a faulty quantum instrument reconstruction can be found in Fig. 14 of Appendix 8.

Figure 4

Bloch vectors after undergoing QND detection in the no-loss case for different loss channels. (a) State vector evolution for asymmetric loss from ${|0⟩}_q$ is captured by the color gradient, ranging from $0\mathrm{\%}$ loss (bright points) to 100% (dark points) for various input states (i) ${|0⟩}_q$, (ii) ${|1⟩}_q$, (iii) ${|-⟩}_{X,q}$, and (iv) ${|-⟩}_{Y,q}$. Notably, the Bloch vectors remain close to the surface of the sphere, independent of the loss probability; see the Appendix 8. The initial superposition states ${|-⟩}_{X,q}$ and ${|-⟩}_{Y,q}$ are found transitioning to the basis state not affected by the loss. (b) The erasure channel is realized by consecutively inducing the same partial loss from ${|0⟩}_q$ followed by ${|1⟩}_q$ and postselecting to both no-loss cases, i.e., both ancilla’s ${|0⟩}_a$ outcome. Our results support the theory derivation of a map $\propto \rho$ leaving the initial states up to noise unaltered; see the Appendix 8.

Figure 5

Tomographic reconstruction of the maps characterizing our quantum instrument. (a) Single-qubit Choi operators in the elementary basis $\{{|00⟩}_q,\dots ,{|11⟩}_q\}$ describing the QND detection under loss from ${|0⟩}_q$. Process fidelities compared to the ideal map for $2\mathrm{\%}$ and $61\mathrm{\%}$ losses read $0.97(1)$ and $0.98(1)$, respectively. Black boxes denote the ideal operator in the higher loss case. (b) On the erasure channel we receive the expected identity map for loss from both qubit states up to about 60% before errors start to dominate. (c) Corresponding process fidelities compared with ideal maps together with the decay model (dashed line) from Eq. (7).

Figure 6

Multiqubit entangled state undergoing QND detection in the no-loss case. As an input, we choose the four-qubit GHZ state $(1/\sqrt{2})({|0000⟩}_q+{|1111⟩}_q)$. Loss is induced from ${|0⟩}_{q,1}$ on system qubit 1. Results for purity ($◯$) and population ratio between the GHZ basis states $|0000⟩$ and $|1111⟩$ ($\mathrm{♢}$) in analogy to the Bloch-vector picture (Fig. 4) are shown. The purity is found constant, while the population ratio increases towards higher loss probabilities, finally causing a distortion to state ${|1111⟩}_q$ not affected by the loss. Errors correspond to one standard deviation of statistical uncertainty due to quantum projection noise.

Figure 7

Full-system dynamics from combined qutrit-ancilla quantum instrument tomography. (a) Choi operator of the system qutrit evolution in the elementary basis $\{{|00⟩}_q,\dots ,{|22⟩}_q\}$ after postselecting on the ancilla, revealing either loss case (rows) examined for different loss probabilities from ${|0⟩}_q$ (columns). The tricolor Choi operators show peaks in the absence of loss (blue), peaks occurring due to partial loss (orange), and erroneous peaks (red). The latter are only color coded on the diagonal for visualization purposes. Process fidelities with the ideal map from top left to bottom right read $\{0.97(1),0.96(1),0.95(1),0.83(1),0.86(1),0.84(1)\}$. (b) False-positive and false-negative rates extracted from raw data for loss states $\{{|0⟩}_q,{|1⟩}_q,1/\sqrt{2}({|0⟩}_q+{|1⟩}_q)\}$ versus the loss probability.

Figure 8

Simulations on the faulty QND loss detection embedded in the seven-qubit color code. (a) A single logical qubit encoded on a triangular planar color code lattice formed of three interconnected plaquettes (lower part). The code space is formed by six stabilizer operators $S_x^{(i)}$ and $S_z^{(i)}$, each acting on a plaquette of four physical qubits [67]. Loss is subsequently detected on all code qubits using a faulty QND circuit (top part). We model this taking into account both correlated and single-qubit over-rotations, representing our leading error mechanisms by treating every qubit as a qutrit. (b) Single QEC cycle of qubit loss detection and correction, including initial controlled induction of loss, followed by faulty QND loss detection operations on the qubit subspace of all physical qutrits and stabilizer measurements triggering respective conditional Pauli corrections.

Figure 9

Logical error rates simulated for a loss correction cycle of the seven-qubit color code with faulty stabilizer measurements. The logical error rates are shown as a function of the loss probability $p_{\mathrm{loss}}$ induced by the QND-detection scheme of Fig. 8 for different error rates $q$ in the stabilizer readout. The black line (with equation $1-p_{\mathrm{loss}}$) represents the error rate when no encoding is performed. The logical error rates for the ideal case with no over-rotation errors in the QND loss detection unit are shown with green up-pointing triangles. Blue circles show the logical error rates when the QND-detection unit is simulated with over-rotation parameters ($p_{\mathrm{corr}}=0.045$, resulting in a stabilizer measurement error rate $q=0.045$ and $p_{\mathrm{single}}=2.47\times {10}^{-4}$) coming from the experimental data. Data simulated with $q=p_{\mathrm{corr}}=0.023$ corresponding to an improvement in the $R^{\mathrm{MS}}$-gate fidelity is shown with orange down-pointing triangles. In the region with $0.03\lesssim p_{\mathrm{loss}}\lesssim 0.33$, error correction is beneficial in protecting the logical states with respect to storing information in an unencoded single physical qubit.

Figure 10

Comparison between the coherent and incoherent implementations of the faulty QND loss detection unit. (a),(b). Logical error rates for $p_{\mathrm{loss}}=0$ as a function of (a) the single-qubit over-rotation rate $p_{\mathrm{single}}$ for $p_{\mathrm{corr}}=0$ and (b) the correlated over-rotation rate $p_{\mathrm{corr}}$ for $p_{\mathrm{single}}=0$ after a round of error correction of the seven-qubit color code following the scheme in Fig. 8 where the imperfections in the QND loss detection unit are implemented either with a coherent or an incoherent noise channel. (c) Logical error rate as a function of the loss probability when the faulty QND loss detection unit is modeled as a coherent channel. Panel (d) is the same as (c), but when errors in the QND loss detection are modeled as an incoherent Clifford channel.

Figure 11

Investigating the performance of the QND-detection unit according to Fig. 2 in the main text. Population in the $D_{5/2}$ state for the system qubit (directly measured loss) versus transferred excitation on the ancilla qubit (detected loss) in the case of detecting loss repeatedly using both ancilla qubits $a_1$ and $a_2$ (left), solely with ancilla $a_1$ (middle), and ancilla $a_2$ (right). The imprinted detection efficiencies demonstrate reliable loss mapping onto the ancilla qubit and its readout by means of QND. Errors correspond to one standard deviation of statistical uncertainty due to quantum projection noise.

Figure 12

Purity of a single system qubit after undergoing QND detection in the no-loss case for different loss channels. (a) After a single QND detection with loss from ${|0⟩}_q$, we find purity values unaffected by the amount of loss for all of the given input states. At high loss probabilities, tomography becomes unreliable due to the low count rates. Furthermore, the superposition states show systematic drifts beyond the given statistical errors, which however do not affect the results. (b) The erasure channel is realized by consecutively inducing the same partial from ${|0⟩}_q$ followed by ${|1⟩}_q$ and postselecting to both ancilla ${|0⟩}_a$ outcomes. The purity of the output state is again unaffected by loss for any of the probed input states. Errors correspond to one standard deviation of statistical uncertainty due to quantum projection noise.

Figure 13

Correlations between two repeated QND detections according to Fig. 2 in the main text. Loss on the system qubit is induced from the imprinted states followed by two repeated detections using ancillae $a_1$ and $a_2$. A positive correlation refers to successfully detecting the same loss event twice, whereas faulty assignments can be separated into false-positive and false-negative events; shown in the lower figure part. Errors correspond to one standard deviation of statistical uncertainty due to quantum projection noise.

Figure 14

Potential risks on system qubit process reconstruction when partial tracing the ancilla qubit. Left column: partial tracing before reconstructing the map directly on the raw data, previously used in Fig. 7 of the main part. In this case, the loss level ${|2⟩}_q$ is incoherently added onto state ${|1⟩}_q$. Coherences present in ${|01⟩}_q$ originate from the reconstruction technique, forcing physical properties. Right column: postselecting from the already reconstructed qubit-qutrit maps presented in Fig. 15. In contrast to before coherences on ${|01⟩}_q$ vanish, whereas the nonphysical bias remains.

Figure 15

Combined process reconstruction on the ancilla-qutrit system according to Fig. 16. The resulting Choi operators (right column) denoted in the elementary basis ($\{{|0000⟩}_{a,q},\dots ,{|1212⟩}_{a,q}\}$) describe the whole dynamics of the QND-detection unit under loss from ${|0⟩}_q$. Hue relates to phase according to the top right color bar and saturation to the absolute value of the complex entries. Process fidelities with the ideal Choi operators (left column) from top to bottom read $\{0.91(1),0.89(1),0.85(1)\}$. Errors correspond to one standard deviation of statistical uncertainty due to quantum projection noise.

Figure 16

Schematics on higher-dimensional process tomography. (a) Qutrit process tomography solely covering the system qubit $(q)$ together with the loss level $\{{|0⟩}_q,{|1⟩}_q,{|2⟩}_q\}$ undergoing the QND-detection unit by using nine preparation settings together with six measurement settings, resulting in 54 experiments each run. (b) Combined process tomography on ancilla $(a)$ and qutrit $(1)$, capturing the entire dynamics of this quantum instrument using 12 settings on the ancilla qubit (four preparation settings and three measurement settings) alongside 54 settings on the system qutrit, resulting in 648 experiments. (c) Qutrit process tomography on the erasure channel, focusing on the no-loss case, i.e., twice postselecting the ancilla qubit’s ${|0⟩}_a$ outcome.

Figure 17

Qutrit process tomography characterizing the QND-detection unit for loss from ${|0⟩}_q$ according to Fig. 16. (a) System qutrit’s Choi operator in elementary basis $\{{|00⟩}_q,\dots ,{|22⟩}_q\}$ after tracing over the ancilla qubit and various loss probabilities denoting the effect of the loss transition transferring population from ${|0⟩}_q$ to ${|2⟩}_q$. (b) The respective fidelities with the ideal Choi operators covering the complete loss range.

Figure 18

Qutrit process tomography characterizing the QND-detection unit for loss from ${|1⟩}_q$ according to Fig. 16. (a) System qutrit’s Choi operator in elementary basis $\{{|00⟩}_q,\dots ,{|22⟩}_q\}$ after tracing over the ancilla qubit and several different loss probabilities denoting the effect of the loss transition transferring population from ${|1⟩}_q$ to ${|2⟩}_q$. (b) The respective fidelities compared to the ideal Choi operators covering the complete loss range.

Figure 19

Qutrit process tomography on two repeated QND detections for loss from ${|0⟩}_q$ according to Fig. 16. (a) System qutrit Choi operators mapping loss repeatedly onto ancillae $a_1$ and $a_2$ under several different loss probabilities. The processes for which we trace over both ancillae prior to reconstruction denote the loss transition transferring population from ${|0⟩}_q$ to ${|2⟩}_q$. (b) Fidelities compared to the ideal operators remain approximately constant along all measured loss probabilities and show slightly decreased values compared to the results on single QND detection from Fig. 17.

Figure 20

Noise-model QND detection. Various noise model describing the experimental limitations on the ancilla-qutrit Choi operator depicted in Fig. 15. The limitations are best described when combining correlated coherent rotations together with depolarizing and dephasing noise. Correlated errors clearly dominate as depolarizing and dephasing errors only lead to minor improvements. The error parameters on the bottom of each plot refer to depolarizing error $p_{\text{depol.}}$, dephasing error $p_{\text{deph.}}$, and correlated error $p_{\text{corr.}}$, the latter according to Eq. (A15). Lines connect the points for clarity.

Figure 21

Comparison between experimental and noisy-modeled Choi operators. (a) Experimentally estimated map according to Fig. 15 with additionally marked transitions denoting correlated errors describing our leading error mechanism; see Eq. (A15). (b) Most suitable noisy-modeled Choi operator combining correlated, depolarizing, and dephasing errors.

Figure 22

Comparison between the coherent and incoherent implementations of the faulty QND loss detection unit in the case of no losses. (a) Logical error rate as a function of the correlated over-rotation rate $p_1$ for the parameter $p_2=0.045$ obtained from the experimental data. (b) Logical error rate as a function of the correlated over-rotation rate $p_2$ for the parameter $p_1=2.47\times {10}^{-4}$ obtained from the experimental model.