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A generative modeling approach for benchmarking and training shallow quantum circuits
Open accessAbstract
Hybrid quantum-classical algorithms provide ways to use noisy intermediate-scale quantum computers for practical applications. Expanding the portfolio of such techniques, we propose a quantum circuit learning algorithm that can be used to assist the characterization of quantum devices and to train shallow circuits for generative tasks. The procedure leverages quantum hardware capabilities to its fullest extent by using native gates and their qubit connectivity. We demonstrate that our approach can learn an optimal preparation of the Greenberger-Horne-Zeilinger states, also known as “cat states”. We further demonstrate that our approach can efficiently prepare approximate representations of coherent thermal states, wave functions that encode Boltzmann probabilities in their amplitudes. Finally, complementing proposals to characterize the power or usefulness of near-term quantum devices, such as IBM’s quantum volume, we provide a new hardware-independent metric called the qBAS score. It is based on the performance yield in a specific sampling task on one of the canonical machine learning data sets known as Bars and Stripes. We show how entanglement is a key ingredient in encoding the patterns of this data set; an ideal benchmark for testing hardware starting at four qubits and up. We provide experimental results and evaluation of this metric to probe the trade off between several architectural circuit designs and circuit depths on an ion-trap quantum computer.
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TL;DR
The paper proposes data-driven quantum circuit learning (DDQCL), a hybrid quantum-classical generative modeling framework that uses Born-rule probabilities from shallow quantum circuits and introduces the qBAS score for benchmarking NISQ/hybrid systems. It demonstrates learning GHZ/cat-state preparations, coherent thermal states, and Bars-and-Stripes patterns in simulations and on an ion-trap quantum computer.
Key findings
- DDQCL learned GHZ-like state preparation recipes from classical samples: 3- and 5-qubit instances yielded one recipe, while 4- and 6-qubit instances yielded another; parameters converged close to π/2, and the recipes generalized to larger odd and even qubit numbers.
- For coherent thermal states with N = 5 qubits, deeper circuits (L = 3) consistently outperformed shallower circuits (L = 2 and L = 1) as the task became harder by lowering temperature from 2T_c to T_c to T_c/1.5; the inverse Bethe approximation performed extremely well in the easy task, matched L = 3 in the intermediate task, and underperformed in the difficult task.
- For BAS(2, 2), the minimum entanglement entropy of any state encoding the data set is S_BAS(2,2) = 1.25163 and the maximum is S_BAS(2,2) = 1.79248; DDQCL found a representation with S_BAS(2,2) = 1.69989. All-to-all topology outperformed sparse topologies, and qBAS(2, 2) was computed with N_reads = 15 samples.
- The qBAS score is an F1-based, hardware-independent metric on Bars and Stripes; example readout requirements include N_reads = 15 for (2, 2), 120 for (4, 4), and 16,780 for (10, 10).
Methods
- DDQCL maps binary data vectors to computational-basis states of an N-qubit circuit. The circuit uses alternating layers of arbitrary single-qubit rotations and Mølmer-Sørensen XX entangling gates; Born’s rule gives P_θ(x) = |⟨x|ψ(θ)⟩|². Training minimizes a clipped negative log-likelihood C_nll(θ) = −(1/D) Σ ln(max(ε, P_θ(x^(d)))) using gradient-free particle swarm optimization, with 1000 measurements per estimation.
- Data sets are synthetic: zero-temperature ferromagnet/GHZ, random thermal Boltzmann distributions with N = 5 qubits (and N = 6 in supplementary), and BAS(2, 2). Numerical experiments use 1000 data points sampled exactly from these distributions, 25 random initializations, and bootstrapping for error bars. Simulations used QuTiP2; experiments ran on an ion-trap quantum computer at the University of Maryland.
- The qBAS(n, m) score is the harmonic mean of precision and recall on BAS patterns, with N_reads = N_BAS(n,m) H_{N_BAS(n,m)}. The paper compares all-to-all, chain, and star entangling topologies and different layer counts.
Limitations
- The paper states that the negative log-likelihood cost becomes expensive to estimate as system size increases, that alternative cost functions are only preliminary, that optimizing N_reads is left for future work, and that KL divergence may become undefined for large n × m. It also notes simulations assumed noiseless hardware, deeper circuits did not always improve performance due to optimization/local optima, and the experimental L = 4 improvement was smaller than theoretical because the quantum computer accumulated errors in the deeper circuit.
- Evident from the text: experimental validation is limited to small instances—GHZ training up to 6 qubits, thermal states at N = 5, and BAS(2, 2) on 4 qubits—and qBAS experiments were performed on a single ion-trap device. The GHZ recipe extraction also relied on human expert inspection and rounding of learned parameters.
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41 Own interpretation · 23 Own result · 12 No basis given · 10 Not checked yet · 1 Supported so far
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“The recipes obtained are summarized in Fig. 3 and were verified for larger number of qubits, both odd and even.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
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“Deeper circuits such as L = 3 (purple pentagons) consistently outperformed shallower circuits such as L = 2 (red circles) and L = 1 (yellow triangles).”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
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“However, deeper circuits do not always provide significant improvements, as it is the case for all-to-all L = 4 (dark green circles) versus all-to-all L = 2 (orange circles).”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
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“Finally, it is reassuring that DDQCL obtained circuits for cat state preparation starting from samples of a classical distribution.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
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“This is an important observation, since we know, based on our empirical results, that (i) single layer circuits with no entangling gates severely underperform in producing the output state probability distribution that is close to the target data set”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
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“As shown in Fig. 3, the learning process unveiled that these converge to the same value.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
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“DDQCL produced approximate recipes of this kind in some of the test cases for 3 and 4 qubits when using a single entangling layer with chain topology.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
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“and (ii) when inspecting the parameters learned for circuits with all-to-all topology with L = 2 layers, we found that most of the XX gates reached their maximum entangling setting.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
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“This became more evident as we went from easy learning tasks (Fig. 4a) to hard learning tasks (Fig. 4c).”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
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“We demonstrate that our approach can learn an optimal preparation of the Greenberger-Horne-Zeilinger states, also known as “cat states”.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > Abstract
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In the paper
“For example, learning probabilistic generative models is in many cases an intractable task, and quantum-assisted algorithms have been proposed for both gate-based 17,18 and quantum annealing devices.[19][20][21][22][23]”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > INTRODUCTION
What the sources say
- Not checked yetcites [17] Nathan Wiebe, K. M. S. & Kapoor, A. Quantum deep learning. Preprint at arXiv:1412.3489 (2015).
- Supportedcites [18] Kieferová, M. & Wiebe, N. Tomography and generative training with quantum boltzmann machines. Phys. Rev. A 96, 062327 (2017). source
In “Tomography and Generative Data Modeling via Quantum Boltzmann Training” (p. 3), the paper says the training objective is unlikely to be generically computable because calculating a required term is #P-hard. This supports the qualified claim that learning probabilistic generative models can be intractable, rather than claiming that every such model is hard to train. (LLM, conf=0.87)
- Not checked yetcites [19][20][21][22][23]
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“Furthermore, the maximum value that a quantum representation of BAS(2, 2) can reach is S BAS(2,2) = 1.79248, which happens to be the maximum entanglement entropy known for any four-qubit state 56 (see also Fig. S5 and the corresponding Section in the Supplementary Material).”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
What the sources say
- Not checked yetcites [56] Higuchi, A. & Sudbery, A. W. How entangled can two couples get? Phys. Lett. A 273, 213–217 (2000). source
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“Instances of 3 and 5 qubits yielded a recipe, while instances of 4 and 6 qubit yielded another recipe.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
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“The minimum value of entanglement entropy that any such state can have is S BAS(2,2) = 1.25163.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
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In the paper
“As shown in Fig. 4, the inverse Bethe approximation (green bar) performed extremely well in the easy task (a), matched the L = 3 quantum circuit in the intermediate task (b), and underperformed on the difficult task (c).”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
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“The score is sensitive to the depth of the circuit as shown by the performance improvement of L = 4 compared with L = 2 in the star topology.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
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“All the topologies achieve a lower median KL divergence and the confidence intervals shrink.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > Bootstrapping analysis
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“Finally, tapping into the real power of near-term quantum devices and approximate algorithms implementable on them, we designed a task-specific performance estimator based on a canonical data set.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > DISCUSSION
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“The chain topology (purple squares) performs slightly better than the star topology (red stars) even though they have the same number of parameters.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > Bootstrapping analysis
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“To summarize, first, we learned a GHZ state preparation recipe for an ion trap quantum computer.Minimal intervention by a human expert allowed to generalize the recipe to any number of qubits.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > DISCUSSION
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“We note that a more general performance indicator than qBAS (n, m) score is indeed the KL divergence, D KL [P BAS(n,m) |P θ ].”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > The learning pipeline
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“We start by noting that the four-qubit cat state, whose rich entangled nature makes it ideal for studying decoherence and decay of quantum information, 49,51 has entanglement entropy S GHZ = 1.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
What the sources say
- Not checked yetcites [49] Monz, T. et al. 14-qubit entanglement: creation and coherence. Phys. Rev. Lett. 106, 130506 (2011). source
- Not checked yetcites [51] Ozaeta, A. & McMahon, P. L. Decoherence of up to 8-qubit entangled states in a 16-qubit superconducting quantum processor. Quantum Sci. Technol. 4, 025015 (2019). source
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“DDQCL prepares a quantum state without the assumption of an underlying Boltzmann distribution, while the inverse Bethe approximation infers the parameters with such assumption.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
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“In all these cases, the qBAS score can still be computed and the number of measurements N reads necessary for obtaining a robust estimate continues to remain relatively small to be practical for intermediate size n × m.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > The learning pipeline
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“It would be much more difficult to analyze parameters found with noisy hardware, as DDQCL can learn to compensate certain types of noise, e.g., systematic parameter offsets, in non-trivial ways.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
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“Other classical methods based on machine learning and Markov Chain Monte Carlo, such as the Boltzmann machine, 55 could achieve higher accuracy by requiring more computational resources than the inverse Bethe approximation used here.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
What the sources say
- Not checked yetcites [55] Ackley, D. H., Hinton, G. E. & Sejnowski, T. J. A learning algorithm for boltzmann machines. Cogn. Sci. 9, 147–169 (1985). source
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In the paper
“Hybrid quantum-classical algorithms, such as the variational quantum eigensolver 2,3 (VQE) and the quantum approximate optimization algorithm 4,5 (QAOA), provide ways to use NISQ computers for practical applications.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > INTRODUCTION
What the sources say
- Not checked yetcites [2] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 EP (2014). source
- Not checked yetcites [3] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New J. Phys. 18, 023023 (2016). source
- Not checked yetcites [4] Edward Farhi, S. G. & Goldstone, J. A. quantum approximate optimization algorithm. Preprint at arXiv:1411.4028 (2014).
- Not checked yetcites [5] Hadfield, S. et al. From the quantum approximate optimization algorithm to a quantum alternating operator ansatz. Algorithms 12, 34 (2019). source
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“Although optimization tasks offer a great niche of applications, probabilistic tasks involving sampling have most of the potential to prove quantum advantage in the near-term.[14][15][16]”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > INTRODUCTION
What the sources say
- Not checked yetcites [14][15][16]
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“Furthermore, the error in the inverse Bethe approximation is expected to go to zero with system size, and only above the reference temperature T c (see Section “Methods” for details).”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
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- Not checked yetcites Methods
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“In other words, we can slightly reduce the number of parameter without changing the expressive power of the circuit.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > Simulation of quantum circuits in silico
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“Moreover, the score depends on the classical resources, hyperparameters, and various design choices, making it a good choice for the assessment of the hybrid system as a whole.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > INTRODUCTION
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In the paper
“Hence, any miscalibration or environmental noise will affect this single performance number, enabling comparison between different devices or across different generations of the same device.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > INTRODUCTION
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In the paper
“Similarly, one may use the score to compare classical resources of the hybrid system (e.g., different optimizers).”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
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“Once the number of layers and topology of entangling gates is fixed, the quantum circuits described above provide a template; by adjusting the parameters we can implement a small subset of the unitaries that are in principle allowed in the Hilbert space.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > Gradient-free optimization
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“Previous implementations of Born machines [25][26][27][28] often relied on the construction of tensor networks and their efficient manipulation through graphical-processing units.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > INTRODUCTION
What the sources say
- Not checked yetcites [25][26][27][28]
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The total number of parameters per entangling layer depends on the chosen topology: all is a fully-connected graph and has N(N -1)/2 parameters, chain is a one-dimensional nearest neighbor graph with N -1 parameters, and star is a star-shaped graph with N-1 parameters.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > Simulation of quantum circuits in silico
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In this work, we presented a data-driven quantum circuit learning algorithm (DDQCL) as a framework that can assist in the characterization of NISQ devices and to implement simple generative models.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > DISCUSSION
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In the paper
“Although we compared circuits implemented on the same ion trap hardware, the score may be used to compare different device generations or even completely different architectures (e.g., superconductor-based versus atomic-based).”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
hedgedconditional- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.
In the paper
“Bars and stripes BAS 47 is a canonical machine learning data set for testing generative models.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
What the sources say
- Not checked yetcites [47] MacKay, D. J. C. Information Theory, Inference and Learning Algorithms. (Cambridge University Press, New York, 2002).
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“The only way to reproduce the zero-temperature ferromagnet distribution in our setting is to implement a unitary transformation that prepares a GHZ-like state.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Errors in the device will affect this single performance measure, the qBAS score, which can be used to compare different device generations, or completely different architectures.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > DISCUSSION
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“Note that the number of measurements needed to obtain an unbiased estimate of the relevant probabilities may not scale favorably with the number of qubits N.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > The learning pipeline
hedgedconditional- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
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In the paper
“This is not an example of compilation, but rather an illustration of how simple classical probability distributions can guide the synthesis of interesting non-trivial quantum states.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > DISCUSSION
plainSingle case- Asserted: passes — The authors put it forward as true.
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In the paper
“The zero-temperature ferromagnet distribution is equivalent to assigning 1/2 probability to both |0…0〉 and |1…1〉 states of the computational basis.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > GHZ state preparation
plain- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In contrast, our work does not assume a Boltzmann distribution and therefore does not require a specific sampling beyond Born's rule.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > INTRODUCTION
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“The qBAS score can also be used to benchmark the typical performance of optimizers used in hybrid quantum-classical systems.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > DISCUSSION
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Depending on the level or type of noise in the system, the same algorithm could lead to a different circuit fulfilling the same probability distribution as that of the data.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > DISCUSSION
hedgeddirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The bars and stripes data is easy to generate, visualize and verify classically, while modeling it still requires significant quantum resources in the form of entanglement.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > DISCUSSION
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper may give no reason why it holds. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Yet, implementing a shallow circuit that can uniformly sample such data is hard; we will show that some candidate solutions require large amount of entanglement.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > INTRODUCTION
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“Our approach has the bidirectional capability of using NISQ devices for machine learning, and machine learning for the characterization of NISQ devices.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > DISCUSSION
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper may give no reason why it holds. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In this paper, we propose the qBAS score, a complementary metric designed for benchmarking hybrid quantum-classical systems.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > INTRODUCTION
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In this paper, we used the negative log-likelihood which quickly becomes expensive to estimate as the size of the system increases.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > DISCUSSION
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Finally, complementing proposals to characterize the power or usefulness of near-term quantum devices, such as IBM's quantum volume, we provide a new hardware-independent metric called the qBAS score.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > Abstract
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper may give no reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The total number of parameters per entangling layer depends on the chosen topology: all is a fully-connected graph and has N(N -1)/2 parameters, chain is a one-dimensional nearest neighbor graph with N -1 parameters, and star is a star-shaped graph with N-1 parameters.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > Simulation of quantum circuits in silico
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The score can be computed in any of the NISQ architectures available to date.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > DISCUSSION
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“DDQCL is a modular framework and its performance will ultimately depend on the choices made for such modules.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > DISCUSSION
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The total number of parameters per entangling layer depends on the chosen topology: all is a fully-connected graph and has N(N -1)/2 parameters, chain is a one-dimensional nearest neighbor graph with N -1 parameters, and star is a star-shaped graph with N-1 parameters.”
§ A generative modeling approach for benchmarking and training shallow quantum circuits > Simulation of quantum circuits in silico
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
The authors state this as an assumption.
The data vectors in the data set D are independent and identically distributed random vectors.
“The data set D ¼ ðx ð1Þ ; ; x ðDÞ Þ is a collection of D independent and identically distributed random vectors.”
The authors state this as an assumption.
For setting N_reads, the model distribution is equal to the target distribution P_BAS(n,m).
“Assuming a perfect model distribution P BAS(n,m) = 1/ N BAS(n,m) , the expected number of measurements to obtain a value of r = 1.0 can be estimated using the famous coupon collector's problem.”
The authors state this as an assumption.
For simulations in silico, perfect gate fidelities and error-free measurements are assumed.
“For the simulations in silico, we also assume perfect gate fidelities and error-free measurements.”
The paper relies on this without support.
In the current ion trap experimental setup, arbitrary single-qubit rotations and Mølmer-Sørensen XX entangling gates involving any two qubits can be performed.
“In the current experimental setup, we can perform arbitrary single qubit rotations and Mølmer-Sørensen XX entangling gates involving any two qubits.”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The cited source says what the claim attributes to it.
In the citing paper
“Benedetti et al. [10] proposed the qBAS score metric for benchmarking the shallow quantum-classical hybrid systems using synthetic data set.”
From QASMBench: A Low-Level Quantum Benchmark Suite for NISQ Evaluation and Simulation (2023) · cites this paper as [10]
The check wasn’t confident either way, so it gives no verdict — never disputed.
In the citing paper
“Quantum autoencoders have seen experimental implementation on quantum hardware [168], and will likely be an important primitive in QML.”
From Variational quantum algorithms (2021) · cites this paper as [168]
Experimental — accuracy not yet measured
Citations
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“Figure 2 on the top left panel shows patterns belonging to BAS(2, 2), while the top central panel shows the remaining patterns.”
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Theory for Equivariant Quantum Neural Networks
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Recent Advances for Quantum Neural Networks in Generative Learning
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Quantum machine learning: from physics to software engineering
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Effects of Quantum Noise on Quantum Approximate Optimization Algorithm
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Generation of High-Resolution Handwritten Digits with an Ion-Trap Quantum Computer
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Self-citationExpressive power of parametrized quantum circuits
Du, Y., Hsieh, M.-H., Liu, T., & Tao, D. (2018) · UTS ePRESS (University of Technology Sydney) · Cited by 99
Theoretical guarantees for permutation-equivariant quantum neural networks
Schatzki, L., Larocca, M., Nguyen, Q. T., Sauvage, F., & Cerezo, M. (2024) · npj Quantum Information · doi:10.1038/s41534-024-00804-1 · Cited by 140
A comprehensive review of quantum machine learning: from NISQ to fault tolerance
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On the Quantum versus Classical Learnability of Discrete Distributions
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Adversarial quantum circuit learning for pure state approximation
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Enhancing Generative Models via Quantum Correlations
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Synergistic pretraining of parametrized quantum circuits via tensor networks
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Self-citationQuantum Volume in Practice: What Users Can Expect From NISQ Devices
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From Probabilistic Graphical Models to Generalized Tensor Networks for Supervised Learning
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Benchmarking quantum computers
Proctor, T. J., Young, K. C., Baczewski, A., & Blume-Kohout, R. (2025) · Nature Reviews Physics · doi:10.1038/s42254-024-00796-z · Cited by 116
Classical versus quantum models in machine learning: insights from a finance application
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Self-citationLearning temporal data with a variational quantum recurrent neural network
Takaki, Y., Mitarai, K., Negoro, M., Fujii, K., & Kitagawa, M. (2021) · Physical Review A · doi:10.1103/physreva.103.052414 · Cited by 88
Classification with Quantum Machine Learning: A Survey
Abohashima, Z., Elhosen, M., Houssein, E. H., & Mohamed, W. M. (2020) · arXiv · doi:10.48550/arxiv.2006.12270 · Cited by 68
Linkon, A. A., Shaima, M., Sarker, M. S. U., Badruddowza, B., Nabi, N., Rana, M. N. U., Ghosh, S. K., Esa, H., & Chowdhury, F. R. (2024) · Journal of Computer Science and Technology Studies · doi:10.32996/jcsts.2024.6.1.26 · Cited by 67
Decomposition of matrix product states into shallow quantum circuits
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Self-citationQuantum singular value decomposer
Bravo-Prieto, C., García-Martín, D., & Latorre, J. I. (2020) · Physical Review A · doi:10.1103/physreva.101.062310 · Cited by 64
Information Scrambling in Quantum Neural Networks
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Learning quantum data with the quantum earth mover’s distance
Kiani, B. T., Palma, G. D., Marvian, M., Liu, Z.-W., & Lloyd, S. (2022) · Quantum Science and Technology · doi:10.1088/2058-9565/ac79c9 · Cited by 77
Beyond the Swap Test: Optimal Estimation of Quantum State Overlap
Fanizza, M., Rosati, M., Skotiniotis, M., Calsamiglia, J., & Giovannetti, V. (2020) · Physical Review Letters · doi:10.1103/physrevlett.124.060503 · Cited by 62
The NISQ Analyzer: Automating the Selection of Quantum Computers for Quantum Algorithms
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Generative model benchmarks for superconducting qubits
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Generative Quantum Machine Learning via Denoising Diffusion Probabilistic Models
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Power-optimal, stabilized entangling gate between trapped-ion qubits
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Self-citationAnalyzing the performance of variational quantum factoring on a superconducting quantum processor
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Experimental quantum kernel trick with nuclear spins in a solid
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Best Approximate Quantum Compiling Problems
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Generalized quantum circuit differentiation rules
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Analyzing the Performance of Variational Quantum Factoring on a Superconducting Quantum Processor
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Quantum versus classical generative modelling in finance
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Expressivity of quantum neural networks
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Expressive power of tensor-network factorizations for probabilistic modeling
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Robust resource-efficient quantum variational ansatz through an evolutionary algorithm
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Generative tensor network classification model for supervised machine learning
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Application-Motivated, Holistic Benchmarking of a Full Quantum Computing Stack
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Abrupt transitions in variational quantum circuit training
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Variational Inference with a Quantum Computer
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Variational quantum compiling with double Q-learning
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Style-based quantum generative adversarial networks for Monte Carlo events
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Trainability barriers and opportunities in quantum generative modeling
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Generative quantum learning of joint probability distribution functions
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A Conceptual Architecture for a Quantum-HPC Middleware
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Metadata
- Authors
- Benedetti, M., Garcia-Pintos, D., Perdomo, O., Leyton-Ortega, V., Nam, Y., & Perdomo-Ortiz, A.
- Affiliations
- Cambridge Quantum Computing (United Kingdom) (GB)
- Central Connecticut State University (US)
- IonQ (United States) (US)
- Venue
- npj Quantum Information
- Published
- 2019-05-27
- Volume
- 5
- Issue
- 1
- ISSN
- 2056-6387 (ISSN-L)
- OpenAlex
- W2784994528
- Cited by
- 379
- References
- 71
- Citations by year
- 2026: 42 · 2025: 75 · 2024: 55 · 2023: 49 · 2022: 42 · 2021: 65 · 2020: 29 · 2019: 12
- FWCI
- 16.88
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Locations
- npj Quantum Information (publishedVersion, cc-by)
- arXiv (Cornell University) (submittedVersion)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion)
- UCL Discovery (University College London) (submittedVersion)
- arXiv (Cornell University)
- Funders
- RCUK | Engineering and Physical Sciences Research Council
- Cambridge Quantum Computing Limited
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Keywords
- quantum circuit learning, shallow quantum circuits, generative models, hybrid quantum-classical algorithms, quantum device characterization, GHZ states, Schrödinger cat states, thermal states, quantum volume, entanglement, ion trap quantum computer, qubit connectivity, benchmarking
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Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
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