Abstract
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.
Figures
![FIG. 1. The experimental sample consists of a central λ=4 stub resonator [33], machined out of 6061 aluminum, with a lifetime of 72 μs, 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 λ=2 resonator [34], patterned in the same lithographic step. All four λ=2 resonators have undercoupled input ports for fast individual qubit control. One resonator has a low-Q (1=κ ¼ 60 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 fχig ¼ f1.651; 1.194; 0.811; 0.613g MHz and fχijg 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.](https://s3-eu-west-1.amazonaws.com/com.mendeley.prod.article-extracted-content/images/87316f17-c10c-3d2e-90e2-f7e47c1933fb/thumbnail-e5c37d88-c86f-4198-866e-0d0883ab6747-0.png)
![FIG. 2. Circuit diagram for ZIZ measurement. Steps Xπ refer to one-qubit rotations around the X axis by π radians. Steps X0π indicate that the pulses are spectrally narrow and are roughly selective on having zero photons in the cavity. Steps Di 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 χ1 < χ2 < χ3. 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 displacementD1 to create a coherent state of n̄ ¼ 5 photons into the cavity, which acquires a phase shift θ in a time T ¼ T5 − T0 ≈ θ=ð2πχ1Þ conditionally on the state of qubit 1 (blue line). For measurements of one- and two-qubit properties, θ ¼ 2π=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 T4 − T1 ≈ T=2. The third qubit (red line) would contribute a conditional phase shift of 2πχ3T > θ. We reduce this to θ by performing two Xπ gates separated by T3 − T2 ≈ θðχ−11 − χ−13 Þ=2. At T5, we performD2 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 X0π gate, taking advantage of the well-known number-splitting phenomenon [35]. As the cavity states are separated by ≈6.5 photons, we employ a faster, approximately selective gate, 300 ns in duration. Xπ 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 θ ¼ π, so that the cavity states entangled with the one- and three-excitation manifolds recohere.](https://s3-eu-west-1.amazonaws.com/com.mendeley.prod.article-extracted-content/images/87316f17-c10c-3d2e-90e2-f7e47c1933fb/thumbnail-ad5926c4-4fb0-4f9e-b6c6-2d9c8f8dbb6a-1.png)
![FIG. 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 σi, so that E ¼ P iciσi, 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, σm. Deviations of the identity bar from 0.5 indicate that the meter has some bias in the detectoroutcome distribution. When the amplitude of the σm 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].](https://s3-eu-west-1.amazonaws.com/com.mendeley.prod.article-extracted-content/images/87316f17-c10c-3d2e-90e2-f7e47c1933fb/thumbnail-9d8f1734-3e74-43c4-bef7-9db659310f70-3.png)
![FIG. 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 χ matrix notation: F0ð1ÞðρrÞ ¼ P ijχ 0ð1Þ ij σiρrσj, where fσg are the threequbit 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 Π0ð1Þ ¼ ðIII ZZZÞ=2, and the corresponding χ 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. III F) for this operator to be 80%.](https://s3-eu-west-1.amazonaws.com/com.mendeley.prod.article-extracted-content/images/87316f17-c10c-3d2e-90e2-f7e47c1933fb/thumbnail-ed11606e-4498-408b-bb3a-33f0ea383664-4.png)
Register to see more suggestions
Mendeley helps you to discover research relevant for your work.
Cite
CITATION STYLE
Blumoff, J. Z., Chou, K., Shen, C., Reagor, M., Axline, C., Brierley, R. T., … Schoelkopf, R. J. (2016). Implementing and characterizing precise multiqubit measurements. Physical Review X, 6(3). https://doi.org/10.1103/PhysRevX.6.031041
Readers over time
Readers' Seniority
Readers' Discipline
