Multi-qubit parity measurement in circuit quantum electrodynamics

The essence of error correcting either quantum or classical information is parity checking. In all practical quantum error correction codes [1], the error-free state is signaled by parities of a selection of subsets of qubits all being 'even'; conversely, the occurrence of 'odd' parities indicates a non-trivial error syndrome, with which the particular form of error can be diagnosed. Calculations show that remarkably simple codes are very effective as the substrate of fault tolerant quantum computation; the subsets subjected to parity checks are geometrically local on a two-dimensional (2D) lattice. In addition, the weight of the parity checks is low; for the canonical code of this class, the Kitaev toric code [2], the weights of all checks are four.

It has been standard to assume that a qubit parity should be obtained by computation, in particular by a series of two-qubit quantum logic gates. Thus for the weight-four case, the circuit involves four controlled-NOT gates from each of the four qubits in succession to a fifth, ancilla qubit. The ancilla then holds the value of the parity (or is, perhaps, in an entangled superposition of the two different parity states), so that a measurement of the ancilla reveals (or fixes) this parity. In this paper, we show that, by use of standard microwave scattering techniques, the parity of a small subset of qubits may be measured directly, without the need of an intermediate calculation requiring a logic circuit. We hope that this will simplify the process of error correction, and improve thereby its error robustness.

We first discuss some features of the parity determination that are of a particularly quantum-mechanical character. Firstly, it cannot be trivially assumed that after a quantum measurement records some value, the state of the quantum object necessarily still has that value. For example, if a polarization-sensitive photodetector 'clicks' to indicate the polarization of the photon, the photon possessing that polarization will have vanished. In quantum mechanics there is a name for measurements for which the quantum object remains in the state that is recorded: these are called quantum non-demolition (QND) measurements. Fortunately, there are many implementations of QND-type measurements, and the types of scattering measurements proposed here will have the QND character.

Secondly, there is a choice of basis involved in defining parity. Thus, while in the classical basis (called the 'Z basis' [1]) we would call 0000, 0011, 0110, etc the even parity states, it is possible to take the basis of the qubit to be, for example $|+\rangle =(|0\rangle +|1\rangle )/\sqrt {2}$ and $|-\rangle =(|0\rangle -|1\rangle )/\sqrt {2}$. In this 'X basis', the even parity states are + + + + , + + − − , + − − + , etc. In fact, both the X- and Z-type parity checks are needed in quantum error correction. We will introduce a parity measurement for just one basis, understanding that it is possible to apply one-qubit gates to rotate the qubits so that the parity detection is either of X- or of Z-type. In the superconducting qubit systems that we discuss, these one-qubit rotations can be performed very accurately and quickly.

Thirdly, it is crucial that the parity measurements reveal only the parity, and nothing more. For example, the states 0000 and 0011 should both give parity 'even', and should be seen as identical in the measurement process. This concept has no meaning in the classical setting, where 0000 and 0011 represent objectively different states. Quantum mechanics permits a state like $(|0000\rangle +|0011\rangle )/\sqrt {2}$, which does not have a specific, definite bit state, but which nevertheless has a definite parity. In fact, we employ here a strengthening of the idea of QND, which traditionally requires only that a quantum state remains in a certain subspace after measurement. For the measurements that we need, the state is to remain exactly unchanged, and two states with the same parity should reveal nothing of the differences between them.

In quantum physics, this final concept has been given a name, the quantum eraser [3]. This name refers to the fact that typically in the course of the measurement process, information is temporarily imprinted on the measurement probe, which is however erased by the end of the measurement process. In interferometry this information is the welcher weg ('which path') information which temporarily exists while a photon is moving through the interferometer. This information is erased by the passage of the photon through the final beam splitter of the interferometer. In the implementation we develop below, the measurement probe will temporarily have 'which bitstring' information, which, by the time the scattering process is complete, will be all erased, except for a single bit of parity information. We will show explicitly two different microwave protocols which will permit this quantum eraser condition to be satisfied; we will in fact precisely quantify the degree to which this condition is fulfilled.

Discussions of parity measurements are not new to quantum computing theory; it was understood almost from the beginning that two-qubit parity measurement permitted the implementation of standard two-qubit logic gates [4]. A wide variety of two-qubit implementations have been proposed, in electron optics [5], for spin qubits in quantum dots [6] and for charge qubits [7]. The previous discussions of two-qubit parity measurement for atoms in optical cavities [8], and for superconducting qubits [9, 10], will be particularly relevant for this work. Other theoretical work has simulated the details of the parity measurement process [11, 12], and more generally of 'joint measurement', in which other two-qubit operators than the parity are detected. Such joint measurements have been achieved in the area of circuit QED [13–16], the implementation that we discuss here.

In particular, for the application to fault-tolerant error correction [17, 18], it is very important to go beyond two-qubit parity. While quantum error correction codes are known for which two-qubit parity measurements do suffice [19], they are found to have much worse threshold-rate behavior than the Kitaev code [2], which requires four-qubit checks [20]. The Bacon–Shor codes [21] also permit error correction with only two-bit parity checks, and have rather good error correction performance [22], but cannot achieve fault-tolerant operation by only local operations [23]. Many architectural details of a quantum computer employing the surface code have been worked out [24], and the outlook seems quite favorable [20, 25, 26]. Much less is known about the efficacy of three-qubit parity measurements for error correction, although there is a very promising, recent preliminary result [27].

Extending existing two-qubit parity measurements to more than two is not trivial. Classically combining many two-bit parity results to obtain a multi-qubit parity is not permitted, as the overall quantum eraser condition is not satisfied in this case. A real modification of the measurement protocol or of the coupling structure is necessary. For some other existing schemes [9, 10], we see no reasonable extension to more than two-qubits; in the case of Kerchhoff et al [8] such an extension is possible, as we note below. There is another related proposal for a multi-qubit parity measurement in optical systems [28] based on the acquisition of successive small phase shifts by a probe beam; this scheme seems to require some fine tuning, and has no clear extension beyond three qubits. A very recent proposal shows another technique related to the one we present here for extracting multi-qubit parities [29].

We present here a new solution, requiring a specifically designed cavity (or multi-cavity) structure with multiple, closely spaced resonant modes. We first provide a detailed proposal for performing the three-qubit parity measurement, with accurate satisfaction of the quantum eraser condition, in circuit quantum electrodynamics using superconducting qubits. In the current work the preferred type of superconducting qubit is the 'transmon' type [30], although other types would also be possible [31]. We will always require that the couplings between qubits and microwave resonators be in the well-studied 'dispersive regime' [32, 33], in which the qubit transition frequency and the microwave resonant frequency are well separated. In this regime, the qubit gives a state-dependent shift to the resonant frequency of a microwave cavity—one shift for qubit state |0〉, and a different shift for qubit state |1〉 (note that states 0 and 1 are at different energies). In this regime, the requirements of the multi-qubit parity measurement reduce to those of a classical microwave design problem.

Our first proposal requires a particular hardware arrangement, involving a capability that has been developed only in the most recent experimental literature [34, 35]. In particular, we require the qubits involved in a three-qubit parity measurement each to have equal dispersive coupling to two different resonant modes. In the first version of our proposal described below, these two different modes are realized as the fundamental modes of two different microwave resonators. Crucial for the proposal is that each resonance should occur at nearly (but not exactly) the same frequency. This near-coincidence allows a scattering phase shift to wind through 2 × 2π over a narrow range of frequency. For the four-qubit parity, a winding of 6π is needed, requiring three closely spaced resonant modes to be involved. Our second proposal will deal with this case. We will show that multiple resonances need not require multiple resonators. This proposal will use the currently popular three-dimensional ('3D') cavity [36], but one having a nearly cubical shape so that the three lowest TE₁₀₁-type modes [37] are nearly degenerate. We will show a hypothetical scalable implementation of the surface code within this scheme.

We will first show that the necessary quantum-erasure function can be realized by the choice of circuit hardware illustrated in figure 1. We will work through and discuss the case of the parity measurement for three qubits; generalizing our construction to more qubits is clear, and will be discussed below. In this construction, we have two resonators (we envision 1/4-wave coplanar waveguide resonators [38], as illustrated schematically); in each we will employ one resonant mode, with creation operators a^† and b^†. It would be normal to use the lowest-frequency (fundamental) modes; all other modes are far separated in frequency (the next being at three times the fundamental, in a 1/4-wave structure) and will be ignored. In our scheme, these two fundamental frequencies ω_a and ω_b should be almost, but not exactly, degenerate. The qubit–cavity coupling will be the standard one given by the Jaynes–Cummings model in the dispersive regime [33]. It will be non-standard only in that each qubit will couple to both resonators, so that each qubit j will have a physical coupling strength g_j,a to the a resonator and g_j,b to the b resonator. The Hamiltonian of the system of resonators and qubits can be written as [32]

$$\begin{equation} H=\left(\omega_{a}+\sum_{j=1}^{3}\chi_{a}^{j}\sigma_{z}^{j}\right)a^{\dagger}a+\left(\omega_{b}+\sum_{j=1}^{3}\chi_{b}^{j}\sigma_{z}^{j}\right)b^{\dagger}b+\frac{1}{2}\sum_{j=1}^{3}\omega_{j}\sigma_{z}^{j}. \end{equation} \tag{ 2.1 }$$

Here ω_j are the qubit frequencies, and the dispersive coupling parameters are χ^j_a = g²_j,a/Δ_j,a with Δ_j,a = ω_j − ω_a, and similarly for χ^j_b and Δ_j,b. Note that we assume that the system should be engineered so that there is no direct coupling between qubits.

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As indicated in figure 1, this resonator/qubit structure is to be coupled capacitively to a scattering probe. Rather than extend our Hamiltonian to include all these other details, we proceed in the following way: since H commutes with each σ^j_z, we can examine the Hamiltonian separately in each of its 2³ qubit eigensectors; within each of these sectors H describes a harmonic bosonic system, with qubit-dependent resonant-frequency parameters. Thus, the full scattering experiment can be described quite economically using the classical language of impedance and scattering parameters; the conclusions we draw from this classical discussion will have an immediate, standard quantum interpretation in terms of coherent-state propagation.

From ordinary electrical transmission line theory, the impedance between point A in figure 1 and ground is given by

$$\begin{equation} Z_{A}(\omega)=\mathrm{i}Z_{0}\tan\left(\frac{\pi}{2}\frac{\omega}{\omega_{r,a}}\right). \end{equation} \tag{ 2.2 }$$

Here Z₀ = 50Ω is the impedance of the waveguide. The effective resonant frequency ω_r is dependent on the state of the three qubits $\left |s_{1}s_{2}s_{3}\right \rangle$ according to

$$\begin{equation} \omega_{r,a}=\omega_{a}+\sum_{j=1}^{3}(-1)^{s_{j}}\chi^{j}_a. \end{equation} \tag{ 2.3 }$$

This same discussion applies to the qubit-state-dependent impedance Z_B(ω) of the B resonator. The impedance Z_P (ω) of the entire structure at point P is then given by ordinary series and parallel-combination rules. In fact in the frequency range of interest, the response is very well represented by the lumped circuit of figure 2.

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The measurable quantity for this structure is the reflection coefficient r at P. This is given by [38]

$$\begin{equation} r(\omega)=\frac{V^{-}(\omega)}{V^{+}(\omega)}=\frac{Z_{P}(\omega)-Z_{0}}{Z_{P}(\omega)+Z_{0}}. \end{equation} \tag{ 2.4 }$$

Note that because Z_P is purely imaginary (lossless), |r| = 1, so that only the phase of r,

$$\begin{equation} \theta(\omega)\equiv\arg r(\omega), \end{equation} \tag{ 2.5 }$$

contains information (which can in fact be measured interferometrically).

Our object is to find a probe frequency ω_p such that the reflected signals for all the even-parity qubit states and all the odd-parity states are indistinguishable, but that the reflection coefficients for the even and odd cases are distinct. This will give us conditions on the reflected phase θ_s1s2s3(ω_p) for the different qubit state settings. A general feature of the θ function will make this possible in our two-resonance setting. If ω passes through a resonance of the system (pole of Z_P ), then, while the phase change of the impedance is π, the change of the reflected phase is 2π (cf equations (2.4) and (2.5)). We will arrange that Z_P has two poles within a narrow range of frequency; this means that θ(ω) will vary smoothly over 4π in that range. But, from the point of view of a scattered tone, θ and θ + 2πn are indistinguishable. Thus, the θ(ω) function varies over a sufficient range that we can satisfy our quantum eraser condition for the parity measurement in the following way:

$$\begin{equation} \theta_{\rm even} \equiv \theta_{000}(\omega_{\mathrm{p}})=\theta_{011}(\omega_{\mathrm{p}})+2\pi, \end{equation} \tag{ 2.6 }$$

$$\begin{equation} \theta_{\rm odd} \equiv \theta_{111}(\omega_{\mathrm{p}})=\theta_{001}(\omega_{\mathrm{p}})-2\pi. \end{equation} \tag{ 2.7 }$$

Since the χ coefficient for all qubits is taken to be equal, the other necessary conditions, θ₀₁₁(ω_p) = θ₁₀₁(ω_p) = θ₁₁₀(ω_p) and θ₀₀₁(ω_p) = θ₀₁₀(ω_p) = θ₁₀₀(ω_p), are satisfied automatically. It is also necessary that θ_even ≠ θ_odd (mod 2π), with the best case (most distinguishable) being Δθ ≡ θ_even − θ_odd = π.

One can show that for any choice of parameters in the two-pole circuit, there exists a probe frequency ω_p and a dispersive shift constant χ such that the parity-measurement conditions (equations (2.6) and (2.7)) are satisfied. However, if the resonant frequencies ω_a and ω_b are far apart compared with the width of the resonances, θ_even − θ_odd is very small. By placing the resonances close to one another and choosing the capacitance values carefully, a favorable solution can be found. We note that for further optimization of this structure, there would be no difficulty in replacing the simple pair of capacitors C_a,b with a capacitance bridge in a wye- or delta-configuration [38].

Figure 3 shows a solution for the case of a realistic set of parameters. The choices are ω_a = 2π(9.99 GHz), ω_b = 2π(10.01 GHz), C_a = C_b = 10 fF; note that realistic values for coupling capacitors are in the 0.5 − 50fF range (see [39]). Our equations (2.6) and (2.7) are satisfied (after a simple, efficient numerical search) for ω_p = 2π(9.804 GHz) and χ = 5.77 MHz, giving the nearly optimal value Δθ = 172.9°. Note that the coupling capacitors will introduce T₁ qubit relaxation, but we can estimate that capacitances on this scale give a cavity loss rate of κ ∼ 5 MHz. The Purcell formula for the resulting qubit relaxation time is $T_{1(\mathrm {P})}=\frac {\Delta ^{2}}{\kappa g^{2}}=\frac {\Delta }{\kappa \chi }$. If we assume Δ = 5 GHz, this gives T_1(P) ∼ 200 μs. Thus, the Purcell mechanism for relaxation will not be a severe limit on the lifetime of the qubits.

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We can qualitatively assess the result of applying the measurement tone for a finite length of time. The signal-to-noise ratio for distinguishing even from odd is largely a technical matter involving the noise performance of amplifiers and the effective temperature of filters associated with the resonator-qubit structure. The quantum-erasure property sets a more fundamental limit. If the measurement time is T, the measurement signal will then have a bandwidth W ∼ 1/T around the probe frequency ω_p, because the dispersion of the reflection response is different for the different even and odd states (that is, $\frac {\mathrm {d}\theta }{\mathrm {d}\omega }$ is different for the distinct states, see figure 3). Thus we expect that to maintain the quantum-eraser condition, the bandwidth W should be kept to a small fraction of the resonance width, so perhaps T ∼ 10/χ. This gives a measurement time T ∼ 2 μs. While this is shorter than the expected T₁ in current devices, it would be desirable to shorten T; we expect that further optimization of the scattering structure could make all the $(\frac {\mathrm {d}\theta }{\mathrm {d}\omega })_{\rm odd}$ and $(\frac {\mathrm {d}\theta }{\mathrm {d}\omega })_{\rm even}$ more nearly equal, so that perhaps T could approach 1/χ.

A detailed calculation, given in the appendix, confirms these qualitative considerations. This calculation involves a quantum treatment of the input tone V⁺, in which it is written as a coherent state [40] |α〉, pulsed with a Gaussian time profile with characteristic time T = 1/W. The pulse has mean photon number |α|², and therefore energy ℏω_p|α|². The output tone V⁻ is also a coherent state β _s, but dependent on the qubit state |s〉 = |s₁s₂s₃〉. The coherent state amplitude is always unchanged, |β _s| = |α|, but it is dispersed differently for each state because of the scattering phase shift.

The relevant result from the appendix is

$$\begin{equation} \fl \langle\beta_{{\bf s}}|\beta_{{\bf s'}}\rangle=1-\frac{|\alpha|^{2}b^{2}W^{2}}{2}+O(|\alpha|^{4}b^{4}W^{4}),\,\,\,\,\mbox{qubit parities } {\bf s} \mbox{ and } {\bf s}^\prime \mbox{ the same}, \end{equation} \tag{ 2.8 }$$

$$\begin{equation} \fl \langle\beta_{{\bf s}}|\beta_{{\bf s'}}\rangle=\mathrm{e}^{-|\alpha|^{2}(1-\cos\Delta\theta)}\left(1+O\left(W\frac{\mathrm{d}\theta}{\mathrm{d}\omega}\right)\right),\,\,\,\,\mbox{qubit parities } {\bf s} \mbox{ and } {\bf s}^\prime \mbox{ different}. \end{equation} \tag{ 2.9 }$$

The new parameter b is the first-order difference of phase dispersion

$$\begin{equation} b\equiv\theta'_{{\bf s}}\left(\omega_{\mathrm{p}}\right)-\theta'_{{\bf s'}}\left(\omega_{\mathrm{p}}\right). \end{equation} \tag{ 2.10 }$$

The quantum-eraser condition requires that the 'same parity' cases be indistinguishable (〈β _s|β _s'〉 = 1), and the 'different parity' cases be perfectly distinguishable (〈β _s|β _s'〉 = 0). Noting that, on dimensional grounds, b ∼ 1/χ, these quantum-erasure conditions are well approximated so long as T = 1/W > |α|b ∼ |α|/χ. This confirms the qualitative discussion above, with the additional insight that it is best if the probe is not too strong, i.e. if α is not too large (or course, it should be greater than one to satisfy the 'different parity' condition). If the photon number is taken to be |α|² = 5, and the pulse duration is T ≈ 1 μs, the peak pulse power will be about $P=\frac {\left |\alpha \right |^{2}\hbar \omega _{\mathrm {p}}}{T}=-135\, {\mathrm { dBm}}$. This is indeed a weak signal, but in the detectable range with the current state of the art [41].

2.1. Comparison with Kerchhoff et al

Before presenting our second proposal, we can compare our concept for three-qubit parity with the three-qubit extension of the two-qubit parity measurement that was recently proposed by Kerchhoff et al [8]. They propose sequential scattering from a set of cavities, each coupled to one qubit; the extension of their idea to three qubits, as rendered in plausible microwave components, is shown in figure 4. Note that for n-qubit parity this scheme uses n resonators, while a generalization of our scheme above would use ⌈(n + 1)/2⌉ resonators (see the following section). The Kerchhoff et al scheme is certainly elegant, and simpler in that each qubit only needs to couple to a single cavity. The idea is to choose the coupling and the probe tone such that there is a phase difference of π between qubit states 0 and 1; then this network constructs a sum of the phases for each qubit [42].⁴ Each cavity should have exactly the same resonant frequency (perhaps by tuning). This scheme is more obviously scalable to more qubits than ours, and it would be possible in this scheme to get rid of the first-order dispersion effects which degrade the quantum-eraser condition, as calculated above. But, as we show in the appendix, the second-order dispersion difference, which would still be present, leads to similar qualitative limits on the measurement time and fidelity. Also, we point out that the quantum-eraser condition would also be degraded by imperfections in the circulators, unlike in our scheme. Given that there presently are no on-chip circulators suitable for qubit experiments (but see progress in [43, 44]), we believe that our scheme is closer to being realized with currently available components.

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The theoretical generalization of our scheme to the measurement of the arguably more important case of four-qubit parity is straightforward. We require three closely spaced resonances f_a, f_b and f_c, equal coupling χ of each qubit to each of the three resonances, and a network that couples resonators to a single reflected probe. In this case again all measurable quantities will emerge from a single reflected-phase function θ _s(ω). Since this function will vary smoothly over 6π, it will be possible to satisfy the three quantum-eraser equations for this case:

$$\begin{equation} \theta_{\rm even} = \theta_{0000}(\omega_{\mathrm{p}})=\theta_{0011}(\omega_{\mathrm{p}})+2\pi=\theta_{1111}(\omega_{\mathrm{p}})+4\pi, \end{equation} \tag{ 3.1 }$$

$$\begin{equation} \theta_{\rm odd} = \theta_{0111}(\omega_{\mathrm{p}})=\theta_{0001}(\omega_{\mathrm{p}})-2\pi. \end{equation} \tag{ 3.2 }$$

Tuning the f_i values, χ and ω_p give enough freedom so that these equations should always be solvable. The analogous conditions for n-qubit parity are straightforward to write down; if we use the notation θ_Wt.i to indicate the phase shift if i of the quibits are 1, then the condition is

$$\begin{equation} \theta_{\mbox{w}t.i}(\omega_{\mathrm{p}})=\theta_{\mbox{w}t.i+2k}(\omega_{\mathrm{p}})+2k\pi. \end{equation} \tag{ 3.3 }$$

Combined with the desire that θ_Wt.0(ω_p) ≈ θ_Wt.1(ω_p) + π, we see that the θ function should vary by at least πn. For even n this will be accomplished with n/2 + 1 resonances (so that θ winds through phase π(n + 2)), while for odd n, (n + 1)/2 resonances suffice (θ winds through phase π(n + 1) in this case). For both even and odd n, the number of resonances required can be written ⌈(n + 1)/2⌉, as stated above.

Returning to the consideration of four-qubit parity, we see that some new hardware elements would be required to achieve this by an extension of the '2D' (coplanar waveguide) scheme above. The requirement that each qubit be equally coupled to each of three different coplanar waveguide (CPW) resonators has not previously been achieved. It seems likely that it is doable with the use of air bridges, which have only recently entered the toolkit of quantum microwave engineering (see [45, 46]). Another possible way to obtain the multiple resonant modes needed for multi-qubit parity measurement would involve the use of multi-conductor CPWs [45], which naturally support modes that are closely spaced in frequency.

But given these difficulties, we explore a second protocol to achieve the implementation of four-qubit parity measurement, involving '3D' superconducting cavities [36]. These high-quality factor rectangular cavities have recently been proven to be excellent implementations of nearly decoherence-free Jaynes–Cummings physics [29]. In these the qubit–cavity coupling is provided by antenna structures extending out from the transmon qubits [36]. Multi-qubit structures [47] have been achieved in this technology, and bridging qubits, antenna-coupled to two different cavities, are now possible [48]. This second protocol illustrates a further fact, which is that multiple, closely spaced resonances can be achieved within a single resonant structure.

Figure 5 illustrates the concept of this second protocol, also showing how it could be extended to be part of a scalable implementation of the surface code architecture [24, 49]. The key modification in this structure is that the 3D cavities, rather than having a large aspect ratio in which w ≫ l ≫ h (w = width, l = length, h = height), should be nearly, but not exactly, cubical. Thus, the three lowest resonant modes would be nearly degenerate in frequency, since their wavelengths will be set by π/w, π/l and π/h. (Be warned that the traditional labels for these modes are TE₁₀₁, TE₀₁₁ and TM₁₁₀ [37].) Each cavity is coupled, via antenna structures, to four qubits, two on the front surface in figure 5 and two on the back (not shown). Each antenna runs along the body diagonal (i.e. 〈111〉 axis) of the cubical cavity; with this geometry the coupling is equal to each of the three eigenmodes, whose modal electric field patterns point straight along one of the three coordinate axes.

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Unlike for our first ('2D') proposal, we will not provide calculations of the performance of the structure of figure 5. While the structure of figure 1 can be analyzed using elementary transmission line theory and a few component parameter values whose ranges are well understood, the system of figure 5 is a complex, 3D structure whose electromagnetic response can only be obtained reliably by detailed calculations that are beyond the scope of this paper. It is encouraging that progress is being made in the detailed modeling of couplings in such a 3D geometry [51]. We note that many details of figure 5, such as the thin-film metallization around the qubit, contacts between chip conductors and antenna and cavity metal, the shape of the antennas and the exact geometry of the cavities, should be optimized by detailed simulation. One comment on the cavity structure: while the cubical cavity satisfies the requirement of three closely spaced modes in an elegant way, it will be perhaps discouragingly large (centimeter-scale) from the point of view of potential scale-up of the surface code. Work is commencing on much different forms of resonator geometries, which by being 'quasi-lumped' [52], can be much more physically compact. It will not be necessary to remain only with the platonic cavities.

We consider our basic schemes of figures 1 and 5 to be realistic for implementation by experiment. For figure 1, a precise thin-film layout of an on-chip structure (to the right of point P in figure 1) could be devised based on the parameter values we have determined. It is clear that detailed electromagnetic modeling would be useful to guide the layout design [53]. The values obtained for C_a and C_b correspond to well-known few-finger interdigitated capacitors. We foresee three main difficulties: (i) coupling each qubit to the two resonators and (ii) tuning the qubits to achieve the equal-χ condition. The topology of the chip layout would preclude bringing flux-bias lines to each qubit; it is possible that only two of the three qubits would need to be tuned, and one of these tunings could be from an external magnetic field; (iii) measuring few-photon signal levels. It appears that the standard high electron mobility transistor amplifier arrangements would have too much noise for the conditions we have calculated; adoption of new, quantum-limited amplification will be needed for high-fidelity readout.

The 3D scheme of figure 5 will require more work to assess its optimal implementation. Achieving the parameter values envisioned by our theory will require detailed simulation of the complex structure that we have proposed, with many details subject to variation. Since the direct, ancillaless approach studies here are so different from the existing technique of collecting error syndromes with the action of a quantum circuit using ancillas, we believe that it is impossible to know which approach will be superior. Further studies and refinement of both approaches will be necessary to determine which will provide the best way forward.

So, it may be that our proposals can only attain satisfactory performance in structures in which the resonators are tuneable (as in [54–56]), or qubits are tuneable in frequency (as in [15] and in many other works) or in effective coupling strength (as in [57]). Application of these device construction techniques, as well as near-quantum-limited amplification [41], will be likely needed to achieve high-fidelity, single-shot parity measurement as envisioned in the proposals we give here. We hope that following this route will indeed be facilitated by the many interesting experimental [58–60] and theoretical [12, 61, 62] innovations in the application of circuit QED that we see presently.

The proposals of this paper are not the blueprint for a scalable quantum computer; they are concepts on which further detailed studies to determine optimal device functionality can be based. While the structures that we suggest here will by no means be trivially realized, we believe that the crucial role played by parity measurement in the implementation of reliable quantum computation makes these approaches worthwhile to pursue.

Thanks to Barbara Terhal for emphasizing the importance of multi-qubit parity measurements for fault-tolerant quantum computation; we thank Jay Gambetta, John Smolin and especially Bob W Newcomb, for ongoing discussions. We are grateful for support from the Alexander von Humboldt foundation.

In this appendix, we will compute fidelities between output signals for the case of a finite bandwidth probe.

We compute the fidelity between output signals corresponding to two different qubit states |s〉 and |s'〉. We first construct the creation operator $\hat {a}_{\mathrm { pulse}}^{\dagger }$ corresponding to a finite bandwidth probe pulse as a linear combination of a densely spaced, discrete set of harmonic modes $\hat {a}_{\omega _{i}}^{\dagger }$ [63]:

$$\begin{equation} \hat{a}_{\rm pulse}^{\dagger}=\underset{i}{\sum}C_{i}\hat{a}_{\omega_{i}}^{\dagger} \end{equation} \tag{ A.1 }$$

with

$$\begin{equation} C_{i}=\frac{\sqrt{\delta\omega}\mathrm{e}^{-\left(\omega_{i}-\omega_{\mathrm{p}}\right)^{2}/4W^{2}}}{\left(2\pi W^{2}\right)^{1/4}}, \end{equation} \tag{ A.2 }$$

where ω_p is the center frequency, W is the bandwidth of the probe signal, ω_i is the frequency of mode i and δω is the difference between frequencies of successive modes. In the above expression, the weight C_i of each term is chosen such that in the limit of a continuum of modes (δω → 0) the unitarity constraint is satisfied [63]:

$$\begin{equation} {\sum}\left|C_{i}\right|^{2} \approx \frac{1}{\sqrt{2\pi W^{2}}}\int_{-\infty}^{\infty}\mathrm{d}\omega \,\mathrm{e}^{-\left(\omega-\omega_{\mathrm{p}}\right)^{2}/2W^{2}} \end{equation} \tag{ A.3 }$$

$$\begin{equation} \hspace{45pt}= 1+O(\delta\omega). \end{equation} \tag{ A.4 }$$

Now we define a coherent state of amplitude α for probe pulse:

$$\begin{equation} \left|\alpha\right\rangle =\mathrm{e}^{-\left|\alpha\right|^{2}/2}\,\mathrm{e}^{\alpha\hat{a}_{\rm pulse}^{\dagger}}\left|0\right\rangle . \end{equation} \tag{ A.5 }$$

If we define an amplitude α_i for mode i as follows:

$$\begin{equation} \alpha_{i}\equiv\alpha C_{i}=\alpha\frac{\sqrt{\delta\omega}\,\mathrm{e}^{-\left(\omega_{i}-\omega_{\mathrm{p}}\right)^{2}/4W^{2}}}{\left(2\pi W^{2}\right)^{1/4}} \end{equation} \tag{ A.6 }$$

using the fact

$$\begin{equation} \underset{\delta\omega\rightarrow0}{\lim}\underset{i}{\sum}\left|\alpha_{i}\right|^{2}=\left|\alpha\right|^{2}\underset{\delta\omega\rightarrow0}{\lim}\underset{i}{\sum}\left|C_{i}\right|^{2}=\left|\alpha\right|^{2}, \end{equation} \tag{ A.7 }$$

we can rewrite the coherent probe state $\left |\alpha \right \rangle$ in equation (A.5) in the continuum modes limit as

$$\begin{equation} \left|\alpha\right\rangle =\mathrm{e}^{-\left|\alpha\right|^{2}/2}\,\mathrm{e}^{\alpha\hat{a}_{\rm pulse}^{\dagger}}\left|0\right\rangle \end{equation} \tag{ A.8 }$$

$$\begin{equation} \hspace{18pt}=\mathrm{e}^{-\frac{1}{2}\underset{i}{\sum}\left|\alpha_{i}\right|^{2}}\,\mathrm{e}^{\alpha{\sum}_{i}C_{i}\hat{a}_{\omega_{i}}^{\dagger}}\left|0\right\rangle \end{equation} \tag{ A.9 }$$

$$\begin{equation} \hspace{18pt}=\underset{i}{\prod}\mathrm{e}^{-\left|\alpha_{i}\right|^{2}/2}\,\mathrm{e}^{\alpha_{i}\hat{a}_{\omega_{i}}^{\dagger}}\left|0\right\rangle \end{equation} \tag{ A.10 }$$

$$\begin{equation} \hspace{18pt}=\underset{i}{\prod}\left|\alpha_{i}\right\rangle, \end{equation} \tag{ A.11 }$$

where we have defined a coherent state for each mode i as

$$\begin{equation} \left|\alpha_{i}\right\rangle =\mathrm{e}^{-\left|\alpha_{i}\right|^{2}/2}\,\mathrm{e}^{\alpha_{i}\hat{a}_{\omega_{i}}^{\dagger}}\left|0\right\rangle . \end{equation} \tag{ A.12 }$$

Now if the qubits are in state |s〉 the coherent component $\left |\alpha _{i}\right \rangle$ of the input probe at frequency ω_i will get a phase shift of $\theta _{{\bf{ s}}}\left (\omega _{i}\right )$ and go to the state $\left |\alpha _{i}\,\mathrm {e}^{\mathrm {i}\theta _{{\bf{ s}}}\left (\omega _{i}\right )}\right \rangle$. If we call $\left |\beta _{{\bf{ s}}}\right \rangle$ the state of the output signal when qubits are in state $\left |{\bf{ s}}\right \rangle$ we can compute the fidelity

$$\begin{equation} \mathcal{F} =\left\langle \beta_{{\bf s}}|\beta_{{\bf s'}}\right\rangle \end{equation} \tag{ A.13 }$$

$$\begin{equation} \hspace{12pt}=\underset{i}{\prod}\left\langle \alpha_{i}\,\mathrm{e}^{\mathrm{i}\theta_{{\bf s}}\left(\omega_{i}\right)}|\alpha_{i}\,\mathrm{e}^{\mathrm{i}\theta_{{\bf s'}}\left(\omega_{i}\right)}\right\rangle \end{equation} \tag{ A.14 }$$

$$\begin{equation} \hspace{12pt}=\underset{i}{\prod}\exp\left\{ -\left|\alpha_{i}\right|^{2}(1-\mathrm{e}^{-\mathrm{i}\left(\theta_{{\bf s}}\left(\omega_{i}\right)-\theta_{{\bf s'}}\left(\omega_{i}\right)\right)})\right\} \end{equation} \tag{ A.15 }$$

$$\begin{equation} \hspace{12pt}=\exp\left\{ -\underset{i}{\sum}\left|\alpha_{i}\right|^{2}(1-\mathrm{e}^{-\mathrm{i}\left(\theta_{{\bf s}}\left(\omega_{i}\right)-\theta_{{\bf s'}}\left(\omega_{i}\right)\right)})\right\} . \end{equation} \tag{ A.16 }$$

Expanding the phases around the center frequency of the probe ω_i = ω_p + Δω_i:

$$\begin{equation} \theta_{{\bf s}}\left(\omega_{i}\right) =\theta_{{\bf s}}\left(\omega_{\mathrm{p}}\right)+\theta'_{{\bf s}}\left(\omega_{\mathrm{p}}\right)\Delta\omega_{i}+O\left(\left(\Delta\omega_{i}\right)^{2}\right), \end{equation} \tag{ A.17 }$$

$$\begin{equation} \theta_{{\bf s'}}\left(\omega_{i}\right) =\theta_{{\bf s'}}\left(\omega_{\mathrm{p}}\right)+\theta'_{{\bf s'}}\left(\omega_{\mathrm{p}}\right)\Delta\omega_{i}+O\left(\left(\Delta\omega_{i}\right)^{2}\right). \end{equation} \tag{ A.18 }$$

Now, we consider the case when s and s' have the same parity. Then $\theta _{{\bf{ s}}}(\omega _{\mathrm {p}})=\theta _{{\bf{ s'}}}(\omega _{\mathrm {p}})\mbox{ mod}(2 {\pi})$, and we get to the first order in Δω_i

$$\begin{equation} \mathcal{F}=\exp\left\{ -\underset{i}{\sum}\left|\alpha_{i}\right|^{2}(1-\mathrm{e}^{-\mathrm{i}b\Delta\omega_{i}})\right\}, \end{equation} \tag{ A.19 }$$

where we have defined

$$\begin{equation} b\equiv\theta'_{{\bf s}}\left(\omega_{\mathrm{p}}\right)-\theta'_{{\bf s'}}\left(\omega_{\mathrm{p}}\right). \end{equation} \tag{ A.20 }$$

Using the expression for α_i in equation (A.6) and taking the limit of continuum of modes the sum in the exponent of the above expression becomes an integral

$$\begin{equation} \mathcal{F} =\exp\left\{ -\frac{\left|\alpha\right|^{2}}{\sqrt{2\pi W^{2}}}\int_{-\infty}^{\infty}\mathrm{d}\omega \,\mathrm{e}^{-\left(\omega-\omega_{\mathrm{p}}\right)^{2}/2W^{2}}(1-\mathrm{e}^{-\mathrm{i}b\left(\omega-\omega_{\mathrm{p}}\right)})\right\} \end{equation} \tag{ A.21 }$$

$$\begin{equation} \hspace{10pt}=\exp\left\{ -\left|\alpha\right|^{2}\left(1-\mathrm{e}^{-b^{2}W^{2}/2}\right)\right\} . \end{equation} \tag{ A.22 }$$

If bW ≪ 1 then $1-\mathrm {e}^{-b^{2}W^{2}/2}\simeq \frac {b^{2}W^{2}}{2}$ so that

$$\begin{equation} \mathcal{F}\simeq \mathrm{e}^{-\left|\alpha\right|^{2}b^{2}W^{2}/2}. \end{equation} \tag{ A.23 }$$

If we further assume that $\left |\alpha \right |bW{\ll }1$ we obtain

$$\begin{equation} \mathcal{F}\simeq1-\frac{\left|\alpha\right|^{2}b^{2}W^{2}}{2}. \end{equation} \tag{ A.24 }$$

The fidelity between odd and even states is given by a simpler calculation

$$\begin{equation} \mathcal{F}_{\rm even/odd} =\left\langle \alpha \,\mathrm{e}^{\mathrm{i}\theta_{\rm even}}|\alpha \,\mathrm{e}^{\mathrm{i}\theta_{\rm odd}}\right\rangle \end{equation} \tag{ A.25 }$$

$$\begin{equation} \hspace{42pt} =\mathrm{e}^{-\left|\alpha\right|^{2}(1-\cos\Delta\theta)}\approx \mathrm{e}^{-2\left|\alpha\right|^{2}}. \end{equation} \tag{ A.26 }$$

The final expression is a consequence of the fact that Δθ ≈ π.

Another case we look at, relevant for the alternative scheme of [8], is the case of matching linear dispersion (i.e. b = 0) but finite quadratic dispersion mismatch

$$\begin{equation} b'\equiv\left.\frac{\mathrm{d}^{2}\theta_{{\bf s}}\left(\omega\right)}{\mathrm{d}\omega^{2}}\right|_{\omega_{\mathrm{p}}}-\left.\frac{\mathrm{d}^{2}\theta_{{\bf s}}\left(\omega\right)}{\mathrm{d}\omega^{2}}\right|_{\omega_{\mathrm{p}}}. \end{equation} \tag{ A.27 }$$

Then the fidelity, for the case of the same parity, is

$$\begin{equation} \mathcal{F}=\left|\exp\left\{ -\underset{i}{\sum}\left|\alpha_{i}\right|^{2}(1-\mathrm{e}^{-\mathrm{i}b'\left(\Delta\omega_{i}\right)^{2}})\right\} \right|. \end{equation} \tag{ A.28 }$$

Again in the limit of continuum of modes $\mathcal {F}$ becomes

$$\begin{equation} \mathcal{F} =\left|\exp\left\{ -\frac{\left|\alpha\right|^{2}}{\sqrt{2\pi W^{2}}}\int_{-\infty}^{\infty}\mathrm{d}\omega \,\mathrm{e}^{-\left(\omega-\omega_{\mathrm{p}}\right)^{2}/2W^{2}}(1-\mathrm{e}^{-\mathrm{i}b'\left(\omega_{i}-\omega_{\mathrm{p}}\right)^{2}})\right\} \right| \end{equation} \tag{ A.29 }$$

$$\begin{equation} \hspace{10pt}=\left|\exp\left\{ -\left|\alpha\right|^{2}\left(1-\frac{1}{\sqrt{1+2\mathrm{i}b'W^{2}}}\right)\right\} \right| \end{equation} \tag{ A.30 }$$

$$\begin{equation} \hspace{10pt}=\exp\left\{ Re\left[-\left|\alpha\right|^{2}\left(1-\frac{1}{\sqrt{1+2\mathrm{i}b'W^{2}}}\right)\right]\right\} \end{equation} \tag{ A.31 }$$

$$\begin{equation} \hspace{10pt}=\exp\left\{ -\left|\alpha\right|^{2}\left(1-\sqrt{\frac{1+\sqrt{1+4b'^{2}W^{4}}}{2+8b'^{2}W^{4}}}\right)\right\} \end{equation} \tag{ A.32 }$$

$$\begin{equation} \hspace{10pt} \simeq1-\frac{3\left|\alpha\right|^{2}b'^{2}W^{4}}{2}+O\left(\left|\alpha\right|^{4}\left(b'W^{2}\right)^{4}\right), \end{equation} \tag{ A.33 }$$

where we assumed that b'W² ≪ 1 and $\left |\alpha \right |b'W^{2}{\ll }1$.

To obtain an estimate for the power of the measurement signal, we assume that the signal contains $\left |\alpha \right |^{2}=5$ photons and that it has duration T = 1 μs so that the peak power P will roughly be $P=\frac {\left |\alpha \right |^{2}\hbar \omega _{\mathrm {p}}}{T}=-135\, \mathrm {dBm}$.