Introduction

Deep learning is an established field with pervasive applications ranging from image classification to speech recognition1. Among the most intriguing recent developments is the extension to the quantum regime and the search for advantage based on quantum mechanical effects2. This effort is pursued in a variety of ways, often inspired by the diversity of classical models and based on the concept of artificial neural networks. Prior works have proposed quantum versions of perceptrons3,4, support vector machines5,6, Boltzmann machines7, autoencoders8, and convolutional neural networks9,10,11. The advantage ranges from reducing the model size by exploiting the exponentially large number of amplitudes defining multi-qubit states, to speeding-up either training or inference by applying efficient quantum algorithms such as HHL12 to solve systems of linear equations or reducing the number of samples needed for accurate learning.

A promising implementation is based on variational quantum algorithms in which parametrized quantum circuits are used to prepare approximate solutions to the problem at hand. These solutions are then refined by classically optimizing circuit parameters13. However, fundamental questions must be answered on the parameter landscape14, on the cost of the classical optimization loop, and on the expressive power of circuit ansatze15. Encouraging results suggest that trainability is possible for quantum convolutional neural networks16,17. Still, it is recognized that loading training set data into a quantum machine accurately and efficiently is an unsolved problem18 and, although promising results19,20,21, current solutions work only under specific assumptions. Nevertheless, the exponential complexity of states generated by ever larger quantum computers22 suggests that machine learning techniques will become increasingly important at directly processing large-scale quantum states23.

It was noted in traditional machine-learning literature that non-linear activation functions for neurons are superior24. To translate this observation to the design of quantum neural networks (QNNs), several methods have been proposed to break the intrinsic linearity of quantum mechanics. These solutions range from the use of quantum measurements and dissipative quantum gates25, to the quadratic form of the kinetic term26, reversible circuits27, recurrent neural networks28 and the SWAP test29 with phase estimation30.

Previous work in this context10 has shown the implementation of neural networks applied in post-processing to the classical results of measurements. Here, we experimentally demonstrate a quantum neural network architecture based on variational repeat-until-success (RUS) circuits31,32, that is implemented in a fully coherent way, handling quantum data directly, and in which real-time feedback is used to perform the internal update of neurons. In this model, each artificial neuron is substituted by a single qubit33. The neuron update is achieved with a quantum circuit that generates a non-linear activation function using control-flow feedback based on mid-circuit measurements. This activation function is periodic but locally resembles a sigmoid function. Despite the mid-circuit measurement, this approach does not suffer from the collapse of relevant quantum information. Rather, the measurement outcome signals that the neuron update is either successfully implemented or that a fixed, input-independent operation is performed. This other operation can be undone by feedback and the circuit rerun as necessary until success, leading to a constant, not exponential, overhead in the number of elementary operations required by RUS. Note that the overall fidelity of RUS circuits critically depends on the architecture and speed of the active feedback mechanism.

Our experiment uses 4 of the 7 transmons in a circuit QED processor34 to implement a feedforward QNN with two inputs, one output, and no intermediate layers. We demonstrate that the QNN can learn each of the 16 2-to-1-bit Boolean functions by changing the weights and bias associated with the output neuron. It is particularly noteworthy that this architecture allows implementation of the XOR Boolean function using a single neuron since this is a fundamental example of the limitations of classical artificial neuron constructions, which cannot capture the linear inseparability1 of such a function.

We follow the supervised learning paradigm, in which a set of training examples provides information to the network about the specific function to learn. Our experiment uses multiple copies of a single input state (the maximal superposition of 4 inputs), demonstrating that the QNN can learn from a superposition. Finally, we investigate the specificity of parameters learned for each of the Boolean functions by characterizing how well the values learned for one function can be used for any other. This provides indications on using the QNN to discriminate between the Boolean functions when provided as a quantum black box.

Results

Synthesizing non-linear functions using conditional gearbox circuits

The conditional gearbox circuit35 belongs to a class of RUS circuits31 that use one ancilla qubit QA and mid-circuit measurements to implement a desired operation. The three-qubit version (Fig. 1a) has input qubit QI, output qubit QO, and angles w and b as classical input parameters. For an ideal processor starting with QA in \(\left\vert {0}_{{{{\rm{A}}}}}\right\rangle\), QI in computational state \(\left\vert {k}_{{{{\rm{I}}}}}\right\rangle\) (k ∈ {0, 1}), and QO in the arbitrary state \(\left\vert {\psi }_{{{{\rm{O}}}}}\right\rangle\), the coherent operations produce the state

$$\begin{array}{l}\left\vert {0}_{{{{\rm{A}}}}}\right\rangle \left\vert {k}_{{{{\rm{I}}}}}\right\rangle \left\vert {\psi }_{{{{\rm{O}}}}}\right\rangle \to \sqrt{{p}_{{{{\rm{S}}}}}({\theta }_{k})}\left\vert {0}_{{{{\rm{A}}}}}\right\rangle \left\vert {k}_{{{{\rm{I}}}}}\right\rangle {R}_{x}^{g({\theta }_{k})}\left\vert {\psi }_{{{{\rm{O}}}}}\right\rangle\\ \qquad \qquad\quad \quad\,\,\; + \sqrt{1-{p}_{{{{\rm{S}}}}}({\theta }_{k})}\left\vert {1}_{{{{\rm{A}}}}}\right\rangle \left\vert {k}_{{{{\rm{I}}}}}\right\rangle {R}_{x}^{-\frac{\pi }{2}}\left\vert {\psi }_{{{{\rm{O}}}}}\right\rangle ,\end{array}$$
(1)

where θk = kw + b and \({p}_{{{{\rm{S}}}}}({\theta }_{k})={\cos }^{4}\left(\frac{{\theta }_{k}}{2}\right)+{\sin }^{4}\left(\frac{{\theta }_{k}}{2}\right)\). A measurement of QA in its computational basis produces outcome mA = + 1 (projection to \(\left\vert {0}_{{{{\rm{A}}}}}\right\rangle\)) with probability pS(θk). In this case, the net effect on QO is a rotation around the x axis of its Bloch sphere by angle g(θk), where

$$g({\theta }_{k})=2\arctan \left({\tan }^{2}\left(\frac{{\theta }_{k}}{2}\right)\right)$$
(2)

is a non-linear function with a sigmoid shape (Fig. 1d). This outcome constitutes success.

Fig. 1: Conditional gearbox circuit using repeat until success.
figure 1

a Three-qubit circuit with input parameters (w, b) ideally implementing \({R}_{x}^{g(w+b)}\) on QO for \({{{{\rm{Q}}}}}_{{{{\rm{I}}}}}=\left\vert 1\right\rangle\) and \({R}_{x}^{g(b)}\) for \({{{{\rm{Q}}}}}_{{{{\rm{I}}}}}=\left\vert 0\right\rangle\), heralded by mA = +1 (success). For mA = −1 (failure), the circuit ideally implements \({R}_{x}^{-\frac{\pi }{2}}\) on QO. The probabilistic nature of the circuit is rectified using RUS: in case of failure, QA and QO are first reset (\({R}_{x}^{\pi }\) and \({R}_{x}^{\frac{\pi }{2}}\), respectively), and the circuit re-run. The S† symbol corresponds to \({R}_{z}^{-\frac{\pi }{2}}\). b Compilation into the native gate set after circuit optimization and added error mitigation (two refocusing pulses on QO during QI-QA CZ gates and XY-8 sequence38 during measurement). c Illustration of the ideal action of the conditional gearbox circuit on QO when starting in \(\left\vert {0}_{{{{\rm{O}}}}}\right\rangle\). d Comparison of the ideal g(θ) to a Rabi oscillation of QO, showing the non-linearity of g.

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For failure (i.e., outcome mA = −1 and projection onto \(\left\vert {1}_{{{{\rm{A}}}}}\right\rangle\)), the effect on QO is an x rotation by −π/2, independent of k, w, and b. In this case, the effect of the circuit can be undone using feedback, specifically \({R}_{x}^{\pi }\) and \({R}_{x}^{\frac{\pi }{2}}\) gates on QA and QO, respectively. The circuit can then be re-run with feedback corrections until success is achieved. For an ideal processor, the average number of runs to success, 〈NRTS〉, is bounded by 1 ≤ 〈NRTS〉 = 1/pS(θk) ≤ 2. This bound holds even when QI is initially in a superposition state \(\left\vert {\psi }_{{{{\rm{I}}}}}\right\rangle =\alpha \left\vert {0}_{{{{\rm{I}}}}}\right\rangle +\beta \left\vert {1}_{{{{\rm{I}}}}}\right\rangle\). In this general case, the output state upon success is still a superposition but with potentially different amplitudes:

$$\left\vert {0}_{{{{\rm{A}}}}}\right\rangle \mathop{\sum }\limits_{k=0}^{1}{\alpha }_{k}^{{\prime} }\left\vert {k}_{{{{\rm{I}}}}}\right\rangle {R}_{x}^{g({\theta }_{k})}\left\vert {\psi }_{{{{\rm{O}}}}}\right\rangle .$$
(3)

The probability amplitudes can change, from αk to \(\alpha {\prime}\), depending on the initial \(\left\vert {\psi }_{{{{\rm{I}}}}}\right\rangle\), w, b, and NRTS. This distortion of probability amplitudes can be mitigated using amplitude amplification36, which we do not employ here.

We compile the three-qubit conditional gearbox circuit into the native gate set of our processor (Fig. 1b) and evidence its action after one round using state tomography of QO conditioned on success and failure. Figure 2 shows experimental results when preparing QI (QO) in \(\left\vert {1}_{{{{\rm{I}}}}}\right\rangle \,\)\((\left\vert {0}_{{{{\rm{O}}}}}\right\rangle )\), setting b = 0 and sweeping w, alongside simulation for both an ideal and a noisy processor. Qualitatively, the experimental results reproduce the key features of the ideal circuit: we observe a π-periodic oscillation in pS(w) with minimal value 0.5 at w = π/2, and a sharp variation in ZO from +1 to −1 centered at w = π/2. However, the nonzero ZO components observed for both success and failure indicate that the action on QO for both cases is not purely an x-axis rotation. The noisy simulation captures all key nonidealities observed. This simulation includes nonlinearity in single-qubit microwave driving, cross resonance37 effects between QA and QO, phase errors in CZ gates, readout error in QA, and qubit decoherence [see Supplementary Materials]38.

Fig. 2: Synthesis of non-linear functions using a conditional gearbox circuit.
figure 2

a Probability of success and failure at the first iteration of the conditional gearbox circuit (Fig. 1) as a function of w(b = 0). b, c Pauli components of QO assessed by quantum state tomography conditioned on (b) success and (c) failure, for QI prepared in \(\left\vert 1\right\rangle\). d Purity of QO for success and failure. All panels include experimental results (symbols), ideal simulation (dashed curves), and noisy simulation (solid curves).

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We note the existence of different activation circuits that can implement non-linear activation without the need for RUS strategies. Despite the disadvantages of softer non-linearities implemented by such schemes, they may be viable candidates for implementations of QNNs optimized to noisy processors. We provide a detailed comparison of these schemes in the Supplementary Material.

Control-flow feedback on a programmable superconducting quantum processor

Active feedback is important for many quantum computing applications, including quantum error correction (QEC). Past demonstrations of QEC relied on the storage of measurements without real-time feedback39,40. Moreover, real-time feedback has been demonstrated using data-flow mechanisms, where individual operations are applied conditionally41. In contrast, the implementation of RUS hinges on support for control-flow mechanisms in the control setup (Fig. 3), where the entire sequence of operations has to be assessed and executed, depending on the results of measurements, in real-time.

Fig. 3: Quantum control setup.
figure 3

a Schematic of wiring and control electronics, highlighting critical feedback path between outputs of the quantum processor, the analog-interface devices, controller, and the flux-drive lines. b Timing diagram for the critical feedback path. Hashed regions indicate idling operations for each instrument. Latency concerns the time necessary to ensure synchronicity, since instruments are delayed with respect to the AWGs to account for the difference in propagation/latency time between readout and qubit drive instrumentation.

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In our quantum control architecture, a controller sequences the sets of operations to be performed in real-time, controlling various arbitrary waveform generators (AWG) and digitizers to implement the desired program. Therefore, our implementation of control-flow feedback focuses on this controller and achieves a maximum latency of 160 ns. The latency to complete the full feedback loop of the overall control system (controller, analog-interface devices, and the entire analog chain) was measured to be 980 ns. This represents 3% of the worst coherence time, and sets an upper bound on the efficiency of RUS execution with the quantum processor. Further improvements could be achieved by optimizing the design of our readout (RO) AWG for trigger latency and speeding up the task of digital signal processing within the digitizers.

Note that the critical feedback path consists of the entire readout chain in addition to the slowest instrument, whose latency must also be accounted for before the branching condition is assessed and implemented. In our control setup, the slowest instrument is the Flux AWG, due to the latency introduced by various finite input responses and exponential filters implemented in hardware for the correction of on-chip distortion of control pulses42.

Constructing a QNN using RUS circuits

The characteristic threshold shape of g makes it useful in the context of neural networks: the conditional gearbox circuit can be seen as a non-linear activation function, whose rotations are controlled by the input qubits to mimic the propagation of information between network layers. We use these concepts33 to implement a minimal QNN capable of learning any of the 2-to-1-bit Boolean functions (see Supplementary Materials for their definition and naming convention). These 16 functions can be separated into three categories: two constant functions (NULL and IDENTITY) have the same output for all inputs; 6 balanced functions (e.g., XOR) output 0 for exactly two inputs; and 8 unbalanced functions (e.g. AND) have the same output for exactly three inputs.

The 4-qubit circuit shown in Fig. 4 corresponds to a 3-neuron feedforward network. Two quantum inputs (QI1 and QI2) are initialized in a maximal superposition state. Next, the RUS-based conditional gearbox circuit (now with three input angles w1, w2, and b) performs threshold activation of QO, with no bounds placed on the number of times the conditional circuit is allowed to re-run until success. Following RUS (i.e., QA projected to \(\left\vert {0}_{{{{\rm{A}}}}}\right\rangle\)), QA is reused for training set preparation. Here, the Boolean function f is encoded in a quantum oracle mapping \(\left\vert {k}_{{{{\rm{I1}}}}}\right\rangle \left\vert {l}_{{{{\rm{I2}}}}}\right\rangle \left\vert {0}_{{{{\rm{A}}}}}\right\rangle \to \left\vert {k}_{{{{\rm{I1}}}}}\right\rangle \left\vert {l}_{{{{\rm{I2}}}}}\right\rangle \left\vert f{(k,l)}_{{{{\rm{A}}}}}\right\rangle\). At this point, the 4-qubit register is ideally in state

$$\mathop{\sum }\limits_{k,l=0}^{1}{\alpha_{kl}^{\prime}} \left\vert {k}_{{{{\rm{I1}}}}}\right\rangle \left\vert {l}_{{{{\rm{I2}}}}}\right\rangle \left\vert f{(k,l)}_{{{{\rm{A}}}}}\right\rangle {R}_{x}^{g({\theta }_{kl})}\left\vert {0}_{{{{\rm{O}}}}}\right\rangle ,$$
(4)

where θkl = kw1 + lw2 + b. Finally, QA and QO are compared by mapping their parity onto QA and performing a final measurement on QA on a computational basis.

Fig. 4: Quantum neural network using the repeat-until-success conditional gearbox circuit.
figure 4

a Schematic representation of simplest feedforward network, highlighting the role played by parameters (w1, w2, b) in weighing the sum of input signals, before the result is passed through a non-linear activation function. QI1 and QI2 are input nodes, QO is the output node and QA is an ancilla used first within the RUS circuit and then as expected output for the training set. b Quantum circuit for a 3-neuron feedforward network. This circuit is divided into four steps. Input (QI1, QI2) preparation into maximal superposition; threshold activation into QO using RUS conditional gearbox circuit with (w1, w2, b); unitary encoding of Boolean function (AND, in this case) using oracle; and comparison of QA with QO. XY-4 specifies an error mitigation sequence38 applied during measurement and the symbol ⊔ denotes parking of spectator qubit QI2(QI1) during CZ(QA, QI1) (CZ(QA, QI2)) gates [Supplementary Materials].

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We define C = (1 − mA)/2 from the output mA and estimate \(\langle C\rangle \in \left[0,1\right]\) by averaging over 10,000 repetitions of the full circuit. Training the QNN to learn a specific Boolean function thus amounts to minimizing 〈C〉 over the 3-D input parameter space. Beforehand, we explore the feature space landscapes. Figure 5 shows 2-D slices of 〈C〉 and 〈NRTS〉 for three examples: XOR, IMPLICATION2, and NAND (see Supplementary Materials for slices of all 16 functions). These slices are chosen to include the optimal settings minimizing 〈C〉 for an ideal quantum processor [see Supplementary Materials]. These landscapes exemplify the complexity of the feature space and highlight the various symmetries and local minima that can potentially affect the efficient training of parameters.

Fig. 5: Feature space landscapes of three Boolean functions.
figure 5

2-D slices of 〈C〉 and 〈NRTS〉 for XOR (a, b) IMPLICATION 2 (c, d), and NAND (e, f). For each function, the slice includes (w1, w2, b) parameters that minimize 〈C〉 for an ideal quantum processor. Black dots indicate the experimental parameters achieving minimal〈C〉within each slice.

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Training a QNN from superpositions of data

To train the QNN, we employ an adaptive learning algorithm43 to minimize 〈C〉 over the full 3-D parameter space. Figure 6 shows the training process for NAND, chosen for the complexity of its feature space. The parameters evolve with each training step, starting from a randomly chosen initial point, then exploring the bounds, and subsequently converging to the global minimum in ~50 training steps, requiring ~1 min per step. This satisfactory behavior is observed for all the Boolean functions.

Fig. 6: Learning the NAND function.
figure 6

a Training the QNN to learn NAND over the full parameter space (w1, w2, b) by minimizing 〈C〉 with an adaptive algorithm. Training starts from a randomly-chosen point, then explores the boundaries, and ultimately converges within ~ 50 steps. b–e Evolution of training parameters (w1, w2, b) and 〈C〉 as a function of training step. The current best setting achieved is marked by a star.

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Following training of the QNN for each Boolean function, we investigate the specificity of learned parameters by preparing the 256 pairs of trained parameters and function oracles and measuring〈C〉for each pair. To understand the structure of the experimental specificity matrix (Fig. 7), it is worthwhile to first consider the case of an ideal processor (see Supplementary Materials). Along the diagonal, we expect〈C〉 = 0 for constant and balanced functions, which can be perfectly learned, and〈C〉 ≈ 0.029 for unbalanced functions, which cannot be perfectly learned due to the finite width of the activation function g(θ). In other words, no combination of (w1, w2, b) can lead to perfect contrast for these functions. For off-diagonal terms, we expect〈C〉at or close to multiples of 0.25, the multiple being set by the number of 2-bit inputs for which the paired training function and oracle function have different 1-bit output. For example, NAND and XOR have different outputs only for input 00, while TRANSFER1 and NOT1, which are complementary functions, have different output for all inputs. Note that every constant or balanced function, when compared to any unbalanced function, has a different output for exactly two inputs. Evidently, while the described pattern is discerned in the experimental specificity matrix, deviations result from the compounding of decoherence, gate-calibration, crosstalk, and measurement errors. These errors affect the 256 pairs differently for two main reasons. First, the average circuit depth of the RUS-based conditional gearbox circuit is higher for unbalanced functions. Second, the fixed circuit depth of oracles is also significantly higher for unbalanced functions, as these all require a Toffoli-like gate which we realize using CZ and single-qubit gates. Noisy simulation [Supplementary Materials] modeling the main known sources of error in our processor produces a close match to Fig. 7.

Fig. 7: Specificity of the quantum neural network.
figure 7

Cost function of the optimized parameter set for every training function (horizontal axis) against all oracle functions (vertical axis). In each axis, the functions are ordered from constant, to balanced, to unbalanced. Functions are put alongside their complementary function (NULL and IDENTITY, TRANSFER1 and NOT1, etc.). For an ideal processor,〈C〉values are expected at or close to multiples of 0.25, due to the varying overlap between the 16 Boolean functions (i.e., the number of 2-bit inputs producing different 1-bit outcomes). Further differences arise in the experiment due to variations in the average circuit depth of the RUS-based activation functions and in the fixed circuit depth of oracle functions.

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Despite the evident imperfections, we have shown that it is possible to train the network across all functions, arriving at parameters that individually optimize each landscape. The circuit is thus able to learn different functions using multiple copies of a single training state corresponding to the superposition of all inputs, despite the complexity of feature space landscapes for various Boolean functions.

Discussion

We have seen that RUS is an effective strategy to address the probabilistic nature of the conditional gearbox circuit, allowing the deterministic synthesis of non-linear rotations. Even at the error rates of current superconducting quantum processors, it allowed the implementation of a QNN that reproduced a variety of classical neural network mechanisms while preserving quantum coherence and entanglement. Moreover, we have shown that this QNN architecture could be trained to learn all 2-to-1-bit Boolean functions using superpositions of training data.

This minimal QNN represents a fundamental building block that can be used to build larger QNNs. With larger numbers of qubits, these neurons could form multi-layer feed-forward networks containing hidden layers between inputs and outputs. Beyond feedforward networks, this minimal QNN is amenable to the implementation of various other network architectures, from Hopfield networks to quantum autoencoders33.

Finally, this work highlights the importance of real-time feedback control performed within the qubit coherence time and the quantum-classical interactions governing RUS algorithms. The ability to implement RUS circuits is in itself a useful result, as the active feedback architecture demonstrated is crucial for various other applications of a quantum computer, including active-reset protocols and the synthesis of circuits of shorter depth relative to purely unitary circuit design31, of value in areas such as quantum chemistry. Moreover, recent work into quantum error correction (QEC) highlights the importance of real-time quantum control in protocols for the distillation of magic states or, when coupled to a real-time decoder, the correction of errors. Similarly to real-time feedback, the construction of a real-time decoder that meets the stringent requirements for QEC with superconducting qubits requires application-specific hardware developments that are the focus of ongoing work.