A controllable three-mode cavity

We consider three resonant modes of a cavity in a χ⁽²⁾ medium, with respective mode operators $$ \widehat{a}$$, $$ \widehat{b}$$, and $$ \widehat{c}$$, and frequencies ω_a, ω_b, and ω_c. We neglect any dissipative dynamics until later sections, where we discuss hardware implementations. The cavity is driven by a coherent classical pump with frequency ω_p. We depict it in Fig. 1, in which the pump may be a microwave frequency electric field or an optical drive. By choosing the frequencies to satisfy

Fig. 1

The triply-resonant nonlinear cavity.

The χ⁽²⁾ medium enables the joint control of three modes. We denote the mode operators respectively by $$ \widehat{a}$$, $$ \widehat{b}$$, and $$ \widehat{c}$$. The χ⁽²⁾ medium enables frequency doubling from $$ \widehat{b}$$ to $$ \widehat{a}$$, and a three-way interaction between modes $$ \widehat{b}$$, $$ \widehat{c}$$, and the control field. The control field is either a a classical microwave drive, E_p(t), or b a classical optical drive of envelope p(t). This three-way interaction is effectively a linear interaction between modes $$ \widehat{b}$$ and $$ \widehat{c}$$ that is controlled by the classical drive. The combination of the fixed frequency-doubling interaction and the controlled linear interaction allows extensive control of the joint nonlinear evolution. This evolution conserves the quantity 2n_a + n_b + n_c, in which n_a, n_b, and n_c are the occupation numbers of the respective modes. We also depict the relative values of the frequencies of the three modes; in a, the frequencies of modes $$ \widehat{b}$$ and $$ \widehat{c}$$ are separated by the much smaller frequency of the microwave drive.

Note that the second term in $$ {\widehat{H}}_{\mathrm{nl}}\left(t\right)$$, which is controlled via the amplitude of the pump, is merely a linear coupling between modes $$ \widehat{b}$$ and $$ \widehat{c}$$. This interaction cannot by itself generate a universal set of quantum gates^34,35. It turns out, however, that it can do so when combined with the time-independent frequency-doubling interaction.

We denote the number of photons in the three modes respectively by n_a, n_b, and n_c, and the corresponding operators for the photon number by $$ {\widehat{n}}_a$$, $$ {\widehat{n}}_b$$, and $$ {\widehat{n}}_c$$. Since the Hamiltonian commutes with $$ 2{\widehat{n}}_a+{\widehat{n}}_b+{\widehat{n}}_c$$, the value of that observable is preserved. The Hamiltonian cannot, therefore, mix subspaces defined by different integer values of 2n_a + n_b + n_c. Nevertheless, it does provide complete control within each subspace by virtue of the fact that the repeated commutators of $$ \widehat{a}{\widehat{b}}^{†2}$$ and $$ {\widehat{b}}^{†}\widehat{c}$$ generate a complete Lie algebra for all such subspaces^28,36–38. It is this fact that provides the power of our processing unit.

In general, to implement quantum gates between the three modes, we will need to generate a set of distinct evolutions, one for each of the $$ 2{\widehat{n}}_a+{\widehat{n}}_b+{\widehat{n}}_c=\mathrm{const}$$ subspaces. We can do that with a single control pulse, as for each subspace, there are many choices for p(t) that generate the same unitary operation. We can use numerical search methods to find a control function p(t) that simultaneously generates the required evolution for each of the set of subspaces. Naturally, we wish to find the control that implements a given gate in the shortest time, a challenge solved as described below.

Lastly, the modes of this quantum processor will need to be actively coupled to the waveguides that carry the quantum states to be processed. Otherwise, the process of capturing the content of the waveguides will be too slow due to the necessarily high quality factor of the modes of the processor. To actively couple the cavity modes to the waveguides, we envision using the method given in Heuck et al.²³.

Compiling unitary operations

To find the control pulse p(t) required to implement a given unitary operation, we employ numerical search methods, an approach often referred to as optimal control^39–41. We introduce a parameterization for p(t) as a piecewise-constant signal in which the duration of each interval is variable. This parameterization is essential because the always-on frequency-doubling component of the Hamiltonian necessitates optimizing the length of the pulse. In order to avoid unphysical pulses, we constrain both the duration and amplitude of each interval by the use of sigmoid functions. The full expression for the resulting unitary operation is

Consistently good performance is obtained even with s < 60. This permits the use of standard automatic differentiation tools, without the need for approximations, such as GRAPE⁴⁰. Our parameterization also has the advantage that it does not allow for pathological pulses. Once we have obtained a piecewise constant control function for a given gate, we use GRAPE and standard regularization techniques to smooth out the pulse, ensuring it has both reasonable bandwidth and power. Throughout the optimization, the robustness of the control to calibration errors is verified. The time scale Δτ is shortened until a threshhold is reached at which the control pulse is no longer robust. The above approach to generating control functions, together with a number of symbolic optimizations, will be presented in detail in a related tutorial³⁹.

Examples of programmable gates

Through the use of our implicitly constrained optimal control method, we can perform with high fidelity any gate that keeps 2n_a + n_b + n_c constant. If the length of the control pulse is unconstrained, and dissipation is neglected, we can achieve fidelities arbitrary close to unity. For gates reported here, we constrain the duration of the control pulses as much as possible before reaching unitary fidelities <0.999. In later sections considering hardware implementations, we also describe the effects of dissipation. Here, we describe a number of important unitary operations that fulfill that constraint, some of which are also depicted in Fig. 2. More general unitary operations can be performed by reshuffling the modes of the three-mode processors, as seen in later sections. Given the long cavity lifetimes requires for these operations, reshuffling necessitates rapid catch, and release of photons from and into the connected waveguides, e.g., by using active control as done in Heuck et al.²³.

Fig. 2

The control pulses implementing the three gates that are used to build our error correction circuit.

The top row shows the real and imaginary parts of the control pulses for each gate. The following rows show how the populations of the modes evolve under each gate for a given initial state. The optimizer produces pulses p(t) such that each of the desired transformations leads to constructive interference at the exact same time. Shorter pulses are possible, at the expense of higher power and bandwidth requirements³⁹, up to a point at which the pulse is too short to perform even a single complete oscillation in a subspace defined by an integer value of 2n_a + n_b + n_c.

Throughout the following paragraphs, we will use the notation $$ ∣n_{\mathrm{a}}{n}_{\mathrm{b}}{n}_{\mathrm{c}}⟩$$ to denote a Fock state with n_a, n_b, and n_c photons in modes $$ \widehat{a}$$, $$ \widehat{b}$$, and $$ \widehat{c}$$, respectively.

We begin with the Toffoli Gate, which is a three-qubit non-Clifford gate, distinguished by the fact that together with just the single-qubit Hadamard gate it enables universal quantum circuits^42,43. Of particular relevance for our purposes is the fact that it usually requires six two-qubit CNOT gates to implement^44,45, while our realization requires only a single application of the three-mode processor. We realize the gate in the Hadamard basis (i.e., our gate is a phase gate with two control qubits) for photonic qubits encoded in a single- or dual-rail configuration. In this basis, the Toffoli unitary maps all joint Fock states to themselves except for the state $$ ∣111⟩$$ to which it applies a π phase.

We also define a conditional routing gate as one that swaps the state of two modes depending on the state of a third mode. This class of gates is useful for breaking down conditional multi-qudit operations into smaller units. We first route the target mode to a particular waveguide, based on the state of the control mode, and we perform the appropriate single-mode quantum operation in the new physical location of the target mode. Such routing is indispensable, if our goal is to avoid measurements in error-correcting circuits, as measurements usually require hardware at cryogenic temperatures. Typically, a non-demolition measurement is performed by entangling the required information with an ancilla, and performing a demolition measurement on the ancilla. The result, a classical bit, is then fed forward through a classical computer that decides what quantum operation to perform next. We avoid the measurement and classical decisions through coherent quantum feedback^46,47, where we simply perform a multimode quantum gate conditioned on the ancilla. The realization for the routing gate suggested below is what we use in our bosonic error-correcting circuit, but other setups are feasible as well. Below $$ ∣n_{\mathrm{a}}{n}_{\mathrm{b}}{n}_{\mathrm{c}}⟩$$ denotes a Fock state with n_a, n_b, and n_c photons in modes $$ \widehat{a}$$, $$ \widehat{b}$$, and $$ \widehat{c}$$, respectively. The $$ \widehat{c}$$ mode is the control, the $$ \widehat{b}$$ mode is the input, and $$ \widehat{a}$$ and $$ \widehat{b}$$ are the possible outputs:

When used in the error-correcting circuits described in later sections, mode $$ \widehat{c}$$ will contain an ancillary photon on which routing will be conditioned, while mode $$ \widehat{b}$$ will contain one of the two modes of our multimode bosonic code. A second processor will be used for other modes.

Focusing further on the error-correcting functionality, we need a gate that can correct for photon loss in a codeword. For the code we employ, we require the gate to preserve the states $$ ∣001⟩$$, $$ ∣021⟩$$, and $$ ∣041⟩$$, and accomplish the mapping

This operation is necessary for reverting photon loss in the code mode stored in $$ \widehat{b}$$, while storing information about the occurrence of that loss in mode $$ \widehat{c}$$. Again, we will need two three-mode processors, each acting on one of the modes making up our error-correcting code. Each of the physical modes of the code will be stored in the corresponding $$ \widehat{b}$$ oscillators.

To complete our error correction circuit, we use a gate that entangles two modes. We require this operation because one of the code words is an entangled state, and the loss of a photon breaks this entanglement. This gate provides the mapping

The above gates are only a few of the many operations that the triply-resonant cavity processor can perform. Among these gates are those important for the processing of unprotected single-photon states, and operations that enable unitary modification and number-resolved measurements on modes with more than one photons, including bosonic codes. Importantly, these operations are performed with a single use of the triply-resonant cavity, while otherwise they would require complete circuits with multiple discrete operations. This leads to drastically simpler overall circuits, at the expense of requiring this more sophisticated and difficult to fabricate triply-resonant optical resonator.

Measurement-free error correction

We demonstrate the versatility of our control protocol by constructing an error-correcting circuit around the three-mode processor. The circuit we obtain is not only simple and short, but it also does not require any measurement operations or classical feed-forward control.

We choose the following code, encoding a single qubit in two separate (spatial) bosonic modes, whose logical states are given by

We must first consider the effect of a photon loss on the code. The loss of a photon on the first mode is described by the action of $$ \widehat{a}\widehat{I}$$. This transforms the initial code state $$ ∣C⟩=\alpha {∣0⟩}_{\mathrm{L}}+\beta {∣1⟩}_{\mathrm{L}}$$ into the error state $$ ∣E_{\mathrm{1}}⟩=\alpha ∣12⟩+\beta ∣30⟩$$. Similarly, the loss of a photon from the second mode produces the error state $$ ∣E_{\mathrm{2}}⟩=\alpha ∣21⟩+\beta ∣03⟩$$. For each of these two errors, we need to perform a different correction procedure. Typically, this is achieved by a non-demolition measurement that projects the state of the system onto either the logical subspace or one of the error subspaces, followed by a unitary correction operation conditioned on the measurement result. We sidestep these requirements by using coherent control. We employ two quantum ancillas, initialized to contain single photons, on which routing gates will be conditioned. Thus, our correction procedure involves the following steps. First, we put the information about the presence of an error in the ancillas by using two conditional pumping gates acting in parallel (the code modes are each placed in a $$ \widehat{b}$$ mode, while the ancillary photons are in the corresponding $$ \widehat{c}$$ modes), resulting in the following transformation of the overall ancillas-code state:

Fig. 3

Our minimal architecture for error correction of bosonic codes, readily expandable to larger tasks.

The circuit depicted can be used to correct a single-photon loss using a two-mode bosonic code. The circuit consists of two cavity processors, which for the most part process each mode of the code separately, and a small network of reprogrammable beam splitters and delay lines. These are used to reroute states between the modes of processors as necessary. Each cavity processor is also capable of performing many multi-qubit gates for single- and dual-rail encoded qubits, as well as preparing and manipulating higher-number Fock states. The network of programmable beam splitters between the processors and the delay lines can also be expanded to a fully connected network, enabling universal rerouting between the three modes of each processor for general-purpose quantum computation. The programmable beam splitters can be implemented as Mach–Zehnder interferometers (as shown in the inset) with two 50/50 beam splitters and a programmable delay (the orange medium in the diagram). Classical electronics will be necessary to ensure the pacing of various operations in this device, but no feedback or decision circuitry is necessary, as the approach is measurement free.

Fig. 4

The error-correcting circuit, unrolled in time.

The horizontal axis represents the flow of time, depicting how a pair of triply-resonant cavities is being used. This circuit would be executed on the hardware depicted in Fig. 3. The main drawing is the sequence of operations that we need to perform in parallel in the two triply-resonant cavities, in order to perform the error correction. After placing the code and ancilla modes in the appropriate cavity modes, we accomplish the initial pumping and routing gates. After that, we need to shuffle the ancillary modes by releasing them in the appropriate waveguides. The spatial modes into which the code states are moved depend on the state of the ancillas, thanks to the conditional routing gates. As the ancillas contain information about the presence of photon-loss errors, this lets us perform operations conditioned on the loss of a photon, by performing the two conditional branches in parallel in different physical locations of the circuit. The conditional routing gates then act in reverse, ensuring that all spatial modes end in the same location, without breaking the bijectivity required for any quantum circuit. The various spatial modes employed can be seen in the bottom insets of the figure. Supplementary Figure 1 provides a more detailed rendition. Importantly, as seen in Fig. 3, we do not need 12 triply-resonant cavities as depicted above, rather only 2 cavities with a network of waveguides and programmable beam splitters⁷² that can route the spatial modes as necessary, so that each cavity can be used repeatedly. The gate pictographs are taken from the Font Awesome icon set.

Encoding operation

Encoding a qubit in the two-mode code is particularly simple using the three-mode processor. To do so we have to perform the operation

Two-qubit logical operations

Single-qubit rotations in the logical space of the two-mode code can be realized by using our three-mode processors, as such rotations do preserve 2n_a + n_b + n_c. Moreover, two-qubit logical operations can also be performed. For instance, consider a CPHASE gate, which together with the single-qubit rotations form a universal set. We need to perform the operation

Comparison with other approaches

Comparisons with other codes and types of hardware require care because the various systems have significant differences. Nevertheless, we elucidate how our control protocol substantially reduces the depth of a typical circuit and removes the need for entire classes of expensive operations. As discussed in the introduction, error correction procedures have been proposed for bosonic codes, but these require non-demolition or photon-number-resolving measurements, and it has not yet been described how such measurements can be replaced by unitary operations generated by a χ⁽²⁾ nonlinearity. We can however, compare our circuit to the explicit correction circuit presented in Steinbrecher et al.³³.

One way to compare the efficiency of circuits is to examine how long each takes relative to the characteristic unit of time for the given hardware. The circuit we have constructed above requires six gates, for a total of 40 units of time (relative to the χ⁽²⁾ coupling strength), and four transfers in and out of cavities. The correction circuit employing the quantum optical neural network (QONN) architecture³³, which is the closest analog of our hardware, requires 40 layers, resulting also in 40 units of time, but since it uses a χ⁽³⁾ rather than a χ⁽²⁾ medium, the nonlinearity is significantly weaker, so that the circuit takes longer in real time. Furthermore, the QONN circuit requires 40 transfers in and out of the nonlinear cavities (one for each layer), ten times more than our architecture.

One can instead implement photonic quantum logic by using only the vacuum and one-photon Fock states to encode qubits (i.e., a single- or dual-rail encoding). The smallest error-correcting code in this setting requires five physical qubits²⁶. The logic required to determine the error syndrome for this code requires 16 CNOT gates and 4 auxiliary qubits²⁷. The auxiliary qubits can either be measured, in which case the error can be determined using a classical computer, or a unitary circuit could process the auxiliary qubits and perform the correction^48–52. For each of the 16 different values of the four-bit syndrome, a unitary correction circuit would need to perform a different correction operation. This requires quite a large number of ancillas and CNOT gates, as discussed in the introduction. Our room-temperature design thus represents a dramatic reduction in circuit size and duration. We also emphasize that using all-unitary processes, which is the approach we take here, provides a practical advantage; doing so avoids the need to introduce additional amplification and classical feedback circuitry.

Competing with “active” gate-based approaches to measurement-fee error correction, is the use of continuous autonomous QEC^53–57. In that family of protocols, one needs to design an exotic dissipator, usually through reservoir engineering, which provides an irreversible evolution from the error-space back the code space.

Hardware prospects

We will introduce a less abstract model of our triply-resonant cavity design, in order to better describe the materials science and fabrication challenges it faces. This model also lets us give physical values for the unitless durations we have found above for our control pulses. We will start by describing the physical realization for the $$ \widehat{a}{\widehat{b}}^{†2}$$ and $$ p\left(t\right){\widehat{b}}^{†}\widehat{c}$$ terms in the Hamiltonian. Naturally, these terms requires the presence of eigenmodes $$ \widehat{a}$$, $$ \widehat{b}$$, and $$ \widehat{c}$$. The corresponding field operators would be (e.g., for the $$ \widehat{a}$$ mode)

The coupling rate in this nonlinear Hamiltonian imposes the units of time for the control pulses described in the previous section. This characteristic time needs to be compared to the cavity lifetimes, typically expressed through the Q factor as $$ \tau =\frac{2Q}{{\omega }_{\mathrm{a}}}$$. This lets us introduce the following figure of merit for the characteristic number of operations before the environment destroys our quantum state

Considering some recent SHG on-chip experiments (a Q ~ 10⁷ in refs. ^61,62 and a $$ V_{\mathrm{shg}}~800\mathrm{\mu }{\mathrm{m}}^3~2000\frac{{\lambda }^3}{n^3}$$ with a 70 μm-radius micro-ring in refs. ^63,64, at λ_a ≈ 750 nm) in a typical nonlinear optics material like lithium niobate ($$ {\chi }^{\left(2\right)}~31\frac{\mathrm{pm}}{\mathrm{V}}$$), we obtain values N ~ 0.03, which is still too low for practical use. With Q factors and mode overlaps in SHG experiments following a Moore’s law (the recent progress is explored in Supplementary Note 4) and new designs lowering mode volumes by orders of magnitude^19–21,65, N—the number of elementary quantum operations within the cavity lifetime—could very well grow by orders of magnitude and reach tens to hundreds over the next decade. N is related to a typical figure of merit in SHG experiments—the conversion efficiency⁶⁶ $$ \eta =\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}^2}\propto \frac{Q^3{\left({\chi }^{\left(2\right)}\right)}^2}{V_{\mathrm{shg}}}$$, which has seen impressive improvements in the last decade (see Supplementary Note 4). With a Q ~ 2 × 10⁸, which is achievable in principle⁶⁷, a mode volume of $$ V~10^{-3}\frac{{\lambda }^3}{n^3}$$, (see refs. ^20,68 for progress in mode confinement), and $$ {\chi }^{\left(2\right)}~100\frac{\mathrm{pm}}{\mathrm{V}}$$, which is between the values for lithium niobate and gallium arsenide, we achieve N ~ 2000 which is enough for error correction. Moreover, new fabrication techniques for thin-film materials enable much stronger effective nonlinearities than what has otherwise been achieved on-chip. While such techniques have not been explored extensively in the optical regime, these results are an encouraging indication that similar progress may well be possible for nonlinear optical materials.

To explore how such future hardware may perform, we compare the lifetime of an encoded (protected) photonic qubit to an unprotected single-rail qubit living in the same hardware. The time scale will be set by the Q factor of the cavities under consideration; however, in order to present physical values for the parameters we will set Q ~ 2 × 10⁸ at λ_a ~ 750 nm, which is well within the thermorefractive theoretical limit⁶⁷. In Fig. 5, we compare the performance of our error-correcting protocol to that of an unprotected single-rail qubit, and see that the error-correcting threshold is N ~ 2000, a very demanding value which we are nonetheless optimistic about given the experimental results cited earlier. Typically for non-asymptotic codes, to achieve fault tolerance this lower-level code will have to be concatenated with an asymptotically growing stabilizer code, akin to the surface code or quantum LDPC codes and one of the many techniques for achieving non-Clifford gates (e.g., through magic states) will have to be employed. The versatility of our control protocol provides for a system agnostic to these higher-level architectural decisions.

Fig. 5

Logical qubit lifetime at the “break even” regime where it begins to outperform unprotected qubits.

In blue, we see the decay of a single photon, i.e., an unprotected single-rail qubit. In orange, we see the decay of our two-mode code if we do not perform any correction operations—it decays faster as it contains a higher number of photons. The green line represents the decay of the encoded qubit in the presence of periodic correction operations. The infidelity of the correction operations due to photon loss that can happen during the operation is taken into account. The figure represents a lower bound for the performance of our protocol, with beneficial higher order effects being neglected in order to simplify the simulation. The “break even” point is achieved at $$ V_{\mathrm{shg}}~10^{-3}\frac{{\lambda }^3}{n^3}$$, Q ~ 2 × 10⁸, and $$ {\chi }^{\left(2\right)}~100\frac{\mathrm{pm}}{\mathrm{V}}$$ for λ_a ~ 750 nm. Waveguide losses are neglected, as they would be insignificant compared to the rest of the operations.

Lastly, we need to consider the implementation of the time-dependent control pulses. In the electrical regime, the control pulse can be modulated by standard microwave electronics in CMOS-compatible hardware⁶⁹. In the optical regime, the control pulse would have to be modulated by wave shaping through expressing the pulse in terms of its Fourier decomposition⁷⁰. Intermediate regimes are also possible, in which we can modulate a THz electric field, by placing optically actuated Auston switches next to our triply-resonant cavities⁷¹. Active control will be necessary for loading and unloading photons from these long-lived cavities, e.g., by following methods proposed in Heuck et al.²³.

It is important to note that one can balance the three considerations discussed in this section: the duration, power, and bandwidth of the control pulse. When the values of all these quantities can be expressed in characteristic units close to unity, the optimization problem is well conditioned and easier to solve. Such are the control pulses we have shown (e.g., their amplitudes, bandwidths, and durations are  ≲10). However, if our hardware requires short pulses (e.g., due to low Q factor), but permits high power, we can nudge the solution in this direction by reparameterizing the optimization problem³⁹.