Multiqubit joint measurements in circuit QED: stochastic master equation analysis
 Ben Criger^{1}Email author,
 Alessandro Ciani^{1} and
 David P DiVincenzo^{1, 2}
Received: 3 November 2015
Accepted: 22 March 2016
Published: 4 April 2016
Abstract
We derive a family of stochastic master equations describing homodyne measurement of multiqubit diagonal observables in circuit quantum electrodynamics. In the regime where qubit decay can be neglected, our approach replaces the polaronlike transformation of previous work, which required a lengthy calculation for the physically interesting case of three qubits and two resonator modes. The technique introduced here makes this calculation straightforward and manifestly correct. Using this technique, we are able to show that registers larger than one qubit evolve under a nonMarkovian master equation. We perform numerical simulations of the threequbit, twomode case from previous work, obtaining an average postmeasurement state fidelity of ∼94%, limited by measurementinduced decoherence and dephasing.
Keywords
1 Introduction
Circuit QED provides a promising avenue for the realization of quantum algorithms, with recent experiments showing increases in both coherence time and precision of control [1–3]. Quantum algorithms are thought to require error correction as a prerequisite [4], and quantum error correction requires nondemolition measurement of joint operators, most often Pauli operators of low weight [5–8]. This can be accomplished using an ancilla register which is prepared in a specific state, interacts with the encoded state, and is then measured (possibly destructively) [9–12]. In circuit QED, ancilla measurement has been accomplished by coupling the qubit to photons passing through a resonator, and observing the accrued phase using homodyne detection [13].
Recent work has begun to consider direct joint measurements in circuit QED, in which all qubits in the support of the measured operator are coupled to one or more internal resonator modes, using homodyne detection to observe an output mode, requiring no ancilla qubit. Difficulty in calculating the reduced qubit dynamics has restricted previous analysis of direct measurement schemes to systems containing two [14–16] or three [17, 18] qubits. In this paper, we simplify this calculation, deriving reduced qubit dynamics for an arbitrary number of qubits and resonator modes. We then use the resulting stochastic master equation to extend the analysis of the threequbit, twomode scheme presented in [17, 18].
The rest of this paper is organized as follows. We write the multiqubit, multimode Lindbladian in Section 2, and incorporate it into a stochastic master equation corresponding to homodyne measurement of the output mode. In Sections 3 and 4 we determine the reduced equations of motion for the resonator and register states, respectively. Using these equations, we proceed to simulate multiqubit measurement dynamics in Section 5. We discuss what can be done to increase postmeasurement state fidelity and conclude in Section 6.
2 Parameters
We note that if we use a SchriefferWolff analysis to derive the dispersivecoupling Hamiltonian from an underlying multiqubit JaynesCummings model [19], additional terms appear, which describe qubitqubit and resonatorresonator couplings [20, 21]. In Appendix A we give a full derivation of this expression. We give arguments for why these terms can be neglected (within a rotatingwave approximation) or incorporated into parity measurement schemes with straightforward modifications of the analysis described below. It is also interesting to note that, with a more general starting point provided by circuit Hamiltonians [22], couplings more general than the JaynesCummings form appear, and some qubitqubit and modemode coupling terms can be arranged to cancel, as explored in [22]. We proceed with the model Hamiltonian Eq. (2), as it permits a full and clear exploration of all the issues connected with parity measurement, without the inessential complicating features introduced by the additional coupling terms.
Since, as emphasized by the derivation of Appendix A, the factors \(\lambda_{k,l}\) can have either sign, it is perfectly possible to arrange these factors so that the k sum in the final term of Eq. (3) is zero for every l. In other words, an effective Purcell filter [23] can be created by taking advantage of flexibility provided by the multimode structure. Given the ongoing advances in qubit coherence, we believe it is also reasonable to ignore intrinsic qubit damping, i.e., we can set \(\gamma _{,l}=0\). Thus, from this point onward, we will ignore qubit damping effects (but we will retain qubit dephasing terms).
In the following sections, we simplify the equations of motion further, by deriving the equations of motion for the coherent state amplitudes \(\lbrace\alpha_{k,j} \rbrace\), and incorporating these into the equation of motion for \(\rho_{Q}\), the register reduced state.
3 Resonator equations of motion
3.1 Statespace representation
3.2 Steady states
4 Register equations of motion
4.1 Deterministic
4.1.1 Markovianity
It is interesting to note that the coupling Lindbladian in Eq. (37), though it generates a completelypositive tracepreserving map, is nonMarkovian. This is not surprising, since the Markov approximation is the result of a weakcoupling assumption, and fast quantum measurement requires strong coupling. This has been confirmed in the case of a singlequbit measurement. The coupling Lindbladian, though it can be written in explicit Lindblad form, has a decay rate associated with the dephasing operator which is not necessarily positive [13]. In this section, we prove nonMarkovianity of the coupling Lindbladian in the general case, and we examine the consequences of this property of the Lindbladian by numerical simulation.
Note, however, that if only one qubit is present, the operators \(F_{0}\), \(F_{1}\) and \(F_{2}\) derived above are always linearly dependent. Therefore, the above argument is inapplicable in the onequbit case. However, we observe that the coefficient of the dissipator term of the singlequbit pseudoLindblad equation can be negative in some time intervals during transient evolution, showing that the onequbit evolution also has nonMarkovian features [13]. We expect that nonMarkovianity will be the general case for multimode coupling Lindbladians, since the sum of multiple coefficient matrices with negative eigenvalues is not necessarily positive.
4.2 Stochastic
5 Simulation
In order to assess the accuracy with which joint measurements can be made directly, we focus on the threequbit parity measurement from [17, 18]. We set the resonator parameters according to Eq. (22) (with both \(\kappa_{0}\) and \(\kappa_{1}\) set to 2χ).
5.1 Methods
We simulate the evolution of \(\rho_{Q}\) over the interval \([t,t+dt ]\) in two steps: we first determine the timedependent amplitudes \(\lbrace\alpha_{k,j}(t) \rbrace\) through numerical integration, using a \(4^{\mathrm{th}}\)/\(5^{\mathrm{th}}\)order adaptive RungeKutta stepper, using the pulses detailed in Figure 1. We then use these values of \(\lbrace\alpha_{k,j}(t) \rbrace\) to formulate the timedependent reduced master equation (Eq. (55)), and use an order1.5 stochastic RungeKutta method [25] to integrate it. We repeat this for 10^{5} uniformlyspaced timesteps on the interval \({}[0, \tau ]\), where τ is the total measurement time (taken to be \(\frac {13.5}{\chi}\) throughout).
In order to verify the correctness of the above simulation, we calculate the minimum/maximum eigenvalues, traces, deviations from hermiticity \(( \ \rho \rho^{\dagger} \ _{\infty} )\) and purities (\(\operatorname {tr}(\rho^{2} )\)) for a typical trajectory. In order for these quantities to be meaningful when plotted for a single trajectory, the algorithm used has to be strong (in the terminology of stochastic differential equations [25]). Deviations from hermiticity on the order 10^{−15} are typical, as are deviations from unit trace on the order 10^{−13}, likely caused by numerical rounding error. For the simulations discussed in this paper, the remaining checks are satisfied to within ∼10^{−49}.
5.2 Figure of merit
5.3 Results and discussion
We simulate the action of the measurement in the presence of decoherence from the measurement itself and qubit dephasing noise. Selecting χ as the natural scale for frequency, we take \(\gamma_{z}\) to be \(\frac {\chi}{300}\), as in [18]. If \(\chi \sim1 \times2\pi\mbox{ MHz}\), this would correspond to a dephasing time of ∼50 μs.

unintentional preparation of the \(\vert 1 \rangle\) state on the ancilla qubit,

insertion of a random onequbit Pauli operator (X, Y, or Z) after a memory operation with probability \(\frac {1}{3}\),

insertion of a random twoqubit Pauli operator (IX, IY, …, ZZ) after a CNOT operation with probability \(\frac {1}{15}\), and

an incorrect ancilla measurement outcome
6 Conclusions and future work
The formalism presented in this article facilitates the design of a class of quantum measurement devices, with applications in faulttolerant quantum computing architectures and remote entanglement preparation. Using a pulse with few free parameters, and an approximate matched filter, it is possible to limit transientinduced decoherence, achieving a high state fidelity, comparable to error models studied in the fault tolerance literature operating slightly above the faulttolerance threshold. Further advances in the design of quantum hardware, such as decreases in the dephasing rate, will permit higherfidelity implementations of this protocol. In addition, there are several purely theoretical avenues to be explored, which can inform the feasibility of this idea.
In future work, we will attempt to eliminate or minimize the decoherent portion of the coupling Lindbladian, by selecting a control pulse optimally. Such a control pulse can have a quadraturephase component, unlike the pulse used in this manuscript. Decreasing this decoherence will permit stronger driving, which in turn enables shorter measurement times, reducing the effective strength of intrinsic decoherence. We will also attempt to minimize or correct unwanted rotations, which occur as a result of the acStark shift and the nonzero imaginary part of \(c_{Q}\). It is known that nonlinear filtering of the output photocurrent and multiqubit gates can be used for this purpose [15], but the performance of a scheme involving efficient filters and singlequbit gates has yet to be examined.
7 Remarks
Numerical simulations were performed using the libraries http://github.com/bcriger/homodyne_sim/ and http://github.com/bcriger/sde_solve/, available on Github.
Declarations
Acknowledgements
The authors thank Shabir Barzanjeh, Felix Motzoi, Lars Tornberg, and Anna Vershynina for helpful discussions. The authors acknowledge financial support from ScaleQIT.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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