Implementing and Characterizing Precise Multiqubit Measurements

There are two general requirements to harness the computational power of quantum mechanics: the ability to manipulate the evolution of an isolated system and the ability to faithfully extract information from it. Quantum error correction and simulation often make a more exacting demand: the ability to perform nondestructive measurements of specific correlations within that system. We realize such measurements by employing a protocol adapted from Nigg and Girvin [Phys. Rev. Lett. 110, 243604 (2013)], enabling real-time selection of arbitrary register-wide Pauli operators. Our implementation consists of a simple circuit quantum electrodynamics module of four highly coherent 3D transmon qubits, collectively coupled to a high-$Q$ superconducting microwave cavity. As a demonstration, we enact all seven nontrivial subset-parity measurements on our three-qubit register. For each, we fully characterize the realized measurement by analyzing the detector (observable operators) via quantum detector tomography and by analyzing the quantum backaction via conditioned process tomography. No single quantity completely encapsulates the performance of a measurement, and standard figures of merit have not yet emerged. Accordingly, we consider several new fidelity measures for both the detector and the complete measurement process. We measure all of these quantities and report high fidelities, indicating that we are measuring the desired quantities precisely and that the measurements are highly nondemolition. We further show that both results are improved significantly by an additional error-heralding measurement. The analyses we present here form a useful basis for the future characterization and validation of quantum measurements, anticipating the demands of emerging quantum technologies.

Popular Summary

Measurements play a special role in quantum mechanics and place fundamental limits on what can be known about a quantum system at any one time. Quantum measurement is most often discussed in the context of a two-state system, or qubit, such as the spin of an electron or the polarization of a photon. Developing a quantum computer, however, will require quantum error correction, which calls for subtle measurements of four, five, or even more qubits. Complicating matters further, researchers only want to know the state of qubits with certain properties: For example, is there an odd number of qubits in the $|1⟩$ state? Additionally, scientists wish to determine the quantum state after performing a measurement (i.e., the backaction). Here, we demonstrate an architecture and accompanying protocol uniquely suited to performing these measurements on ensembles of multiple qubits.

Our architecture builds on the success of superconducting three-dimensional transmon qubits, which exhibit excellent coherence times while still allowing high-fidelity control and measurement. We couple four of these qubits together via a high-Q superconducting resonator in such a way that they have negligible cross talk. We then use this resonator to mediate interactions between the qubits prior to directly measuring only one of them. Our algorithm is programmable to measure different properties of our three qubits. We quantitatively characterize each of these measurements, looking at both their accuracy as detectors and the fidelity of the quantum backaction. We distill this analysis into several metrics to benchmark the performance of our measurements.

We expect that our findings will be useful as quantum measurements become increasingly complex.

Figure 1

The experimental sample consists of a central $\lambda /4$ stub resonator [33], machined out of 6061 aluminum, with a lifetime of $72\mu \mathrm{s}$, consistent with our expectation of the limitation due to surface losses. Four sapphire chips enter the cavity radially, each of which supports a 3D transmon and a quasiplanar coaxial $\lambda /2$ resonator [34], patterned in the same lithographic step. All four $\lambda /2$ resonators have undercoupled input ports for fast individual qubit control. One resonator has a low-$Q$ ($1/\kappa =60\mathrm{ns}$) output port that leads to a Josephson parametric converter (JPC) [3]. This enables high-fidelity (98%) readout of the directly coupled qubit, which we designate as the ancilla. The other three qubits are designated as the register, and their associated three resonators are unused. All qubits share essentially identical capacitive geometry, but differing Josephson energies space the qubits by roughly 400 MHz. This results in dispersive shifts $\{{\chi }_i\}=\{1.651,1.194,0.811,0.613\}\mathrm{MHz}$ and $\{{\chi }_{ij}\}$ generally on the order of 1 kHz. Further Hamiltonian and coherence details are in the Supplemental Material [32]. The cavity has an undercoupled input port, used for conditional and unconditional displacements, and a diagnostic output (which is also undercoupled and is not depicted). Panels (a) and (b) depict top view and side view schematics, respectively. Not to scale. (c) False-color top view of the physical device with outlines for clarity.

Figure 2

Circuit diagram for $ZIZ$ measurement. Steps $X_{\pi }$ refer to one-qubit rotations around the $X$ axis by $\pi$ radians. Steps $X_{\pi }^0$ indicate that the pulses are spectrally narrow and are roughly selective on having zero photons in the cavity. Steps $D_i$ represent unconditional displacements of the cavity. The meters are measurements of the ancilla via the readout resonator, which is not itself depicted. The ancilla has the largest dispersive shift and the register qubits are then numerically ordered (from top to bottom) such that ${\chi }_1<{\chi }_2<{\chi }_3$. Prior to this procedure a series of measurements is applied to postselectively prepare the ground state; see the Supplemental Material for details [32]. (a) The algorithm begins with a displacement $D_1$ to create a coherent state of $\overline{n}=5$ photons into the cavity, which acquires a phase shift $\theta$ in a time $T=T_5-T_0\approx \theta /(2\pi {\chi }_1)$ conditionally on the state of qubit 1 (blue line). For measurements of one- and two-qubit properties, $\theta =2\pi /5$. In this example, we perform a full echo on the second qubit (yellow line) by performing two unconditional $X$ gates separated in time by $T_4-T_1\approx T/2$. The third qubit (red line) would contribute a conditional phase shift of $2\pi {\chi }_{3}T>\theta$. We reduce this to $\theta$ by performing two $X_{\pi }$ gates separated by $T_3-T_2\approx \theta ({\chi }_1^{-1}-{\chi }_3^{-1})/2$. At $T_5$, we perform $D_2$ to shift the odd two-parity coherent-state pointer to the zero-photon state. Note that the overlaps between the even two-parity pointer states and the zero-photon state are exponentially suppressed. (b) We map this photon number information onto the ancilla qubit with a $X_{\pi }^0$ gate, taking advantage of the well-known number-splitting phenomenon [35]. As the cavity states are separated by $\approx 6.5$ photons, we employ a faster, approximately selective gate, 300 ns in duration. $X_{\pi }$ gates on the register are centered on this pulse in time to echo away the cavity evolution during this step. (c) To disentangle the cavity pointer states, we essentially invert the pulse sequence of (a), returning the cavity to the vacuum state. We must also echo the ancilla, as it may now be excited. This results in a total gate length of 970 ns. Subsequently, we measure the ancilla qubit. (d) This optional step determines if there are residual photons in the cavity. Since many types of errors result in residual photons, a subsequent photon-number-selective rotation and measurement of the ancilla heralds these errors. When measuring three qubits (e.g., $ZZZ$), we choose $\theta =\pi$, so that the cavity states entangled with the one- and three-excitation manifolds recohere.

Figure 3

Demonstration of parity measurement outcomes. For all seven nontrivial three-qubit subset-parity operators, we show ancilla excitation probabilities as a function of initial register state. In each panel, the three axes specify the initial state of each register qubit, parametrized by rotation angle about the $X$ axis after initialization in the ground state. We depict three plane cuts through that parameter space. The color scale indicates the probability to find the ancilla in the excited state. Panel (a) shows measurement operator ZZZ. The second row (b)–(d) shows two-qubit parity measurements. It is easily seen that the outcome is independent of the preparation of one qubit. The third row (e)–(g) shows single-qubit measurements, reflecting sensitivity to only one preparation axis. We extract assignment fidelities from these data of 89%–95% that improve to 94%–97% with postselection on a success herald.

Figure 4

Results of quantum detector tomography for three selected operators, using the unheralded data sets. We expand the first element $E$ of each POVM in three-qubit generalized Pauli operators ${\sigma }_i$, so that $E=\sum _i{c}_i{\sigma }_i$, and show the magnitudes of the coefficients of that expansion. For measurement of a Pauli operator, each should have two nonzero bars (amplitude 0.5) corresponding to the identity and the operator of the measurement, ${\sigma }_m$. Deviations of the identity bar from 0.5 indicate that the meter has some bias in the detector-outcome distribution. When the amplitude of the ${\sigma }_m$ bar is less than 0.5, it indicates the measurement does not have full contrast along the desired axis. Finite values of the other bars indicate that our measurement has undesired sensitivity to an extraneous property. The POVM $J$ fidelities for the illustrated operators are 95%, 94%, and 91%, respectively. The other four realized measurement operators, as well as reconstructed POVMs from the success-heralded data set, are provided in the Supplemental Material [32].

Figure 5

Three-qubit conditioned quantum process tomography. Experimental quantum process tomography results for (a) even and (b) odd outcome process maps for the three-parity measurement, $ZZZ$. We express our process tomography in the Pauli basis where the conditioned processes can be described using $\chi$ matrix notation: $F^{0(1)}({\rho }_r)=\sum _{ij}{\chi }_{ij}^{0(1)}{\sigma }_i{\rho }_r{\sigma }_j$, where $\{\sigma \}$ are the three-qubit generalized Pauli operators. Here, we show only the corners of these process matrices, all other parts are visually indistinguishable from noise. Data on the full reconstruction, including of the other six measurement operators, are given in the Supplemental Material [32]. The ideal even and odd outcome processes are projectors ${\mathrm{\Pi }}^{0(1)}=(III\pm ZZZ)/2$, and the corresponding $\chi$ matrices have a simple form consisting of only four real components in the generalized Pauli basis, and this ideal form is overlaid with wire frame bars. Note that we plot only the real components as all experimental imaginary components are visually indistinguishable from noise. We calculate the $J$ fidelity (as defined in Sec. 3f) for this operator to be 80%.