Abstract Single-qubit data re-uploading classifiers map every input to a point on the Bloch sphere and are routinely evaluated by thresholding an expectation value estimated from many repetitions of the circuit (shots). A single measurement, however, can deliver at most the Helstrom accuracy of the two class-conditional mixtures, which for one qubit reduces to the length of the difference between the class-mean Bloch vectors. We call the difference between thresholded accuracy and single-shot accuracy the shot gap and measure it in a computational experiment with known ground truth: four synthetic tasks with exact Bayes accuracy, five circuit depths, three training losses, two label-noise levels and ten seeds (1,200 runs), simulated exactly and validated against a state-vector simulation, finite-difference gradients and a brute-force search over measurements. The Helstrom bound held in every run. For circuits of depth two or more, models whose thresholded accuracy averaged 0.857 delivered 0.696 per single measurement, a mean shot gap of 0.161; at depth six, models with 0.93–0.98 thresholded accuracy on the linear, circle and XOR tasks delivered 0.67–0.86 per shot. The embedding, not the choice of measurement, accounted for 96.3% of the single-shot shortfall. Training with the single-shot log-likelihood narrowed the shot gap relative to a smoothed hinge loss in 89% of paired configurations (median reduction 0.085) at a median cost of 0.011 in thresholded accuracy, and eleven shots recovered 74% of the gap against 41% under the hinge loss. Two preregistered hypotheses were not supported: the gap did not grow with boundary nonlinearity, and training loss predicted it better than class-mean Bloch-vector length. Accuracy reports for quantum classifiers should state the number of shots per prediction, as they state training data and compute.
Angle-encoded variational quantum circuits implement truncated Fourier series. For one-dimensional inputs, a single-qubit re-uploading (serial) circuit represents every function that a multi-qubit circuit with the same encoding budget
E can; we find that this exchange of width for depth is not free in first-order training. Because its state lives on the Bloch sphere, all parameters of the serial circuit move the state, at each input, along one complex direction, which we call \emph{phase-locking}. We prove that this costs no Fourier direction but leaves no direction to spare: however many parameters it has, the serial circuit has at most
2E+2 tangent directions, one more than its
2E+1 Fourier coefficients need, while a two-qubit control with the same encoding budget and model class generically has at least
6E+6. The cost in training is measured against this control, as the number of runs out of 100 that reach the optimum within a fixed step budget. On an exact loss with flat-spectrum targets the gap grows with
E; at
E=32 the serial circuit converges in 3, 53 and 83 runs under gradient descent, Adam and Levenberg--Marquardt, the control in 99, 100 and 100. On three real-world series, at matched parameter counts, the serial circuit reaches the least-squares optimum in at most 9 runs under Adam and in 32 to 93 under Levenberg--Marquardt, the control in at least 98 and in all 100; its held-out error is up to about
1.5 times that of the control under Adam, but no larger in the typical run under Levenberg--Marquardt.
Adversarial perturbations threaten machine learning classifiers, including variational quantum classifiers. We show that finite quantum measurement statistics (shot noise) act as a built-in defense against gradient-based test-time attacks whose cost scales unfavorably for the attacker. Because every gradient component must be inferred from repeated circuit executions under any unbiased gradient-estimation rule, white-box extraction consumes a dimension-dependent measurement budget that measurement grouping cannot remove in expressive circuits. Under stated assumptions, single-step attacks need at least quadratically many shots in the input dimension
d, growing as
d5/2 under norm-concentration scaling, with a sufficient-budget analysis for iterative attacks via stochastic gradient Langevin dynamics. Simulations up to 784 input dimensions validate the law: the realized total budget is the
d5/2 geometric floor for plateau-mitigated models and grows as
d3.00 for the tested deep circuits, whose gradient norms decay with dimension absent barren-plateau mitigation; folding the measured gradient norm back in recovers the parameter-free
d3/2 shot-noise geometry. Against a matched classical baseline whose attack overhead is dimension-independent (the cheap-gradient principle of automatic differentiation), the quantum gradient cost ratio grows empirically as
d3.00, so the attacker's relative cost diverges as the model scales. Experiments on a 156-qubit IBM processor (ibm_boston, 4-qubit circuits,
d=12) reproduce the effect: at matched budgets the device attack tracks the ideal within a few percent, with the high-shot gradient faithful to the exact one. The defense operates precisely when the forward map is classically hard to simulate: only then is a white-box attacker denied the simulate-and-backpropagate shortcut and must pay the measurement cost we quantify.
Reporting the Fisher geometry of a trained variational quantum model is routine; quoting the shot budget that would establish it is not. Certifying an empirical Fisher matrix to relative Frobenius error
ε under coordinate-wise parameter shift costs
Θ(Bp2V/(ε2G)) circuit executions, where
V is the measured readout variance and
G the measured squared gradient norm, with uniform allocation optimal in that class. One constant reproduces the cost of two circuit families whose exponents differ by a full power of
p. The exponent is an identity in how
nV and
nG scale with the register, holding family by family to
0.001 across 624 matrix-product-state cells once the finite-
p prefactor is removed. The cubic cost is therefore a finite-size window, set by whether the readout light cone grows with the register. A product family to 256 qubits gives
1.966 (95% CI
1.934--
1.997); a brickwork entangler falls from
2.853 below ten qubits to
1.715 beyond sixty-four; a blocked entangler gives
1.984 at a fixed cone width against
3.034 at a proportional one. Fixed device connectivity fixes the cone, so a cubic budget from a small simulation overestimates a large machine, on top of hardware multipliers
2.07× (
1.41--
3.02),
2.38× and
1.91× on ibm_marrakesh, ibm_fez and ibm_kingston. Cost-optimal readout weights cut the measured shot budget by
2.67× (
1.33--
4.00) on hardware, flat from four to twelve qubits. A discrepancy model fitted on cheap circuits transfers its mean inside the calibration grid and, at six larger sizes named before the data, does not: nominal 90% intervals cover 36%, and split conformal is the only rung that stays near nominal.
Between keeping every shot as a noisy training example and averaging all shots into one clean feature lies an entire spectrum of data organizations that quantum machine learning usually leaves implicit. Shot grouping makes this choice explicit by partitioning a fixed measurement record into disjoint groups and averaging within each group, tuning smoothly between the two conventions with a single validated parameter and no additional quantum executions. We evaluate the method on chaotic synthetic benchmarks and on real-world data under strict execution budgets. At the balanced allocation, intermediate grouping lowers mean error across the evaluated tasks, with supported improvements on every task. Matched controls show that the gain acts as correctly scaled regularization of the classical readout. Unmitigated runs on two superconducting processors show positive mean differences on every task. Shot grouping therefore offers a practical way to improve finite-shot quantum learning at no added quantum cost.
Quantum Machine Learning (QML) has recently emerged as a rapidly growing domain as an intersection of Quantum Computing (QC) and Machine Learning (ML) fields. Hybrid quantum-classical models have demonstrated exponential speedups in various machine learning tasks compared to their classical counterparts. On one hand, training of QML models on real hardware remains a challenge due to long wait queue and the access cost. On the other hand, simulation-based training is not scalable to large QML models due to exponentially growing simulation time. Since the measurement operation converts quantum information to classical binary data, the quantum circuit is executed multiple times (called shots) to obtain the basis state probabilities or qubit expectation values. Higher number of shots worsen the training time of QML models on real hardware and the access cost. Higher number of shots also increase the simulation-based training time. In this paper, we propose shot optimization method for QML models at the expense of minimal impact on model performance. We use classification task as a test case for MNIST and FMNIST datasets using a hybrid quantum-classical QML model. First, we sweep the number of shots for short and full versions of the dataset. We observe that training the full version provides 5-6% higher testing accuracy than short version of dataset with up to 10X higher number of shots for training. Therefore, one can reduce the dataset size to accelerate the training time. Next, we propose adaptive shot allocation on short version dataset to optimize the number of shots over training epochs and evaluate the impact on classification accuracy. We use a (a) linear function where the number of shots reduce linearly with epochs, and (b) step function where the number of shots reduce in step with epochs. We note around 0.01 increase in loss and ~4% (1%) reduction in testing accuracy for reduction in shots by up to 100X (10X) for linear (step) shot function compared to conventional constant shot function for MNIST dataset, and 0.05 increase in loss and ~5-7% (5-7%) reduction in testing accuracy with similar reduction in shots using linear (step) shot function on FMNIST dataset. For comparison, we also use the proposed shot optimization methods to perform ground state energy estimation of different molecules and observe that step function gives the best and most stable ground state energy prediction at 1000X less number of shots.
A single qubit provides sufficient computational capabilities to construct a universal quantum classifier when assisted with a classical subroutine. This fact may be surprising since a single qubit only offers a simple superposition of two states and single-qubit gates only make a rotation in the Bloch sphere. The key ingredient to circumvent these limitations is to allow for multiple data re-uploading . A quantum circuit can then be organized as a series of data re-uploading and single-qubit processing units. Furthermore, both data re-uploading and measurements can accommodate multiple dimensions in the input and several categories in the output, to conform to a universal quantum classifier. The extension of this idea to several qubits enhances the efficiency of the strategy as entanglement expands the superpositions carried along with the classification. Extensive benchmarking on different examples of the single- and multi-qubit quantum classifier validates its ability to describe and classify complex data.