Sampling-based CVaR Variational Quantum Algorithm

• Sampling-based CVaR VQA is a hybrid quantum-classical method that replaces the mean energy objective with a lower-tail risk functional to emphasize elite state sampling.
• The approach uses order statistics by sorting measurement outcomes and averaging the lowest α-fraction, which improves convergence to optimal quantum states.
• Extensions like Ascending-CVaR and smooth tilted-loss surrogates address non-smooth optimization and estimator variance, enhancing algorithm performance under hardware constraints.

Sampling-based Conditional Value-at-Risk variational quantum algorithms are hybrid quantum-classical procedures for combinatorial optimization in which the objective is computed from the lower tail of the energy or loss distribution induced by a parametrized quantum circuit, rather than from its full expectation value. In the standard setting, a circuit $U(\theta)$ prepares $|\psi(\theta)\rangle$, measurements in the computational basis produce bit strings $z$, and a diagonal cost Hamiltonian or directly evaluated classical loss assigns an energy $E(z)$ to each sample. The resulting objective is the lower-tail $\mathrm{CVaR}_\alpha$, estimated from finite shots by sorting measured energies and averaging the smallest $\lceil \alpha K\rceil$ outcomes. This formulation was introduced as an alternative to expectation-based VQE/QAOA for diagonal Hamiltonians (Barkoutsos et al., 2019), and was subsequently extended with evolving risk schedules, direct bitstring-loss evaluation for constrained problems, smooth tilted-loss surrogates, and hardware-scale implementations (Kolotouros et al., 2021).

1. Formal setting and objective

In the diagonal-Hamiltonian formulation, one considers a problem Hamiltonian $H$ acting on $n$ qubits, diagonal in the computational basis with eigenstates $|z\rangle$ for $z\in\{0,1\}^n$ and eigenvalues $|\psi(\theta)\rangle$0. A parametrized state is prepared as

$|\psi(\theta)\rangle$1

and measuring in the computational basis yields $|\psi(\theta)\rangle$2 with probability

$|\psi(\theta)\rangle$3

The induced random variable $|\psi(\theta)\rangle$4 takes values $|\psi(\theta)\rangle$5 with probabilities $|\psi(\theta)\rangle$6, and the standard VQA objective is the expectation

$|\psi(\theta)\rangle$7

Sampling-based CVaR-VQA replaces this mean by a lower-tail risk functional (Barkoutsos et al., 2019).

For minimization, the lower-tail Value-at-Risk and Conditional Value-at-Risk at level $|\psi(\theta)\rangle$8 are

$|\psi(\theta)\rangle$9

and

$z$0

Equivalent integral expressions also appear in the literature, including

$z$1

and, for the energy-distribution setting,

$z$2

At $z$3, CVaR reduces to the expectation; as $z$4, it approaches the minimum supported by the sampled distribution (Barkoutsos et al., 2019).

A second formulation dispenses with an explicit diagonal Hamiltonian and instead evaluates a classical loss directly on each sampled bit string. This pattern is used in constrained portfolio construction and in custom-penalty formulations for constrained binary optimization, where each shot yields a bit string $z$5 and a classically computed value $z$6 or $z$7; the VQA objective is then the empirical $z$8 of those sampled losses (Agliardi et al., 19 Aug 2025). In portfolio applications this point matters conceptually: the VQA objective is CVaR over measured energies from an Ising or directly sampled loss, and is distinct from financial CVaR on portfolio losses (Kolotouros et al., 2021).

2. Sampling estimator and hybrid optimization loop

The defining feature of the sampling-based variant is that the objective is built from order statistics of measurement outcomes. At parameters $z$9, one runs the circuit for $E(z)$0 shots, computes energies or losses $E(z)$1, sorts them as

$E(z)$2

and defines $E(z)$3. The empirical estimator used in the foundational CVaR-VQA work is

$E(z)$4

The same sort-and-average estimator is used in later sampling-based portfolio and constrained-optimization implementations (Barkoutsos et al., 2019).

The hybrid loop is correspondingly simple. One prepares $E(z)$5, samples bit strings, computes a classical energy or loss per shot, sorts the resulting values, feeds the empirical tail average to a classical optimizer, and updates $E(z)$6. The workflow typically also records the best bit string observed so far across all iterations. In the diagonal setting, this best sample is the lowest-energy measured computational basis state; in constrained formulations, it is the lowest classical loss after penalty evaluation (Lee et al., 22 Apr 2026).

Because the objective depends on sorting and on a quantile threshold, the sampled CVaR landscape is non-smooth. The original experiments therefore used gradient-free COBYLA rather than analytic gradients (Barkoutsos et al., 2019). Subsequent studies retained derivative-free optimizers, including COBYLA for VQE/QAOA comparisons (Kolotouros et al., 2021), Powell for step-penalty constrained optimization (Lee et al., 22 Apr 2026), the Nakanishi–Fujii–Todo optimizer for 109-qubit ETF construction (Agliardi et al., 19 Aug 2025), and a two-stage PSO$E(z)$7NFT schedule for 150-qubit dynamic portfolio optimization (Haghighi, 8 Jun 2026).

Sampling overhead is intrinsic to the method. Since only the lowest $E(z)$8-fraction of samples contributes to the objective, the reported analyses note that decreasing $E(z)$9 increases estimator variance and typically requires more shots to match the statistical precision of the mean estimator. One common practical rule is to scale shots by approximately $\mathrm{CVaR}_\alpha$0; in Ascending-CVaR experiments, a baseline $\mathrm{CVaR}_\alpha$1 was scaled as $\mathrm{CVaR}_\alpha$2, and in dynamic portfolio optimization the schedule

$\mathrm{CVaR}_\alpha$3

was used to stabilize tail estimates (Kolotouros et al., 2021). Sorting contributes only an $\mathrm{CVaR}_\alpha$4 classical overhead per objective evaluation (Kolotouros et al., 2021).

3. Landscape reshaping and theoretical properties

The principal theoretical motivation for CVaR-VQA is that, for diagonal Hamiltonians, the practical goal is often not to minimize average measured energy but to increase the probability of sampling low-energy bit strings. CVaR enforces this by discarding or downweighting high-energy outcomes and concentrating the objective on the lower tail. The foundational analysis explicitly states that, for classical optimization problems mapped to diagonal Hamiltonians, aggregating measurement samples by CVaR is more natural than using the expected value (Barkoutsos et al., 2019).

This reshaping can alter the local minima structure. A central proposition in the original work states that a local minimum of $\mathrm{CVaR}_\alpha$5 does not necessarily correspond to a local minimum of $\mathrm{CVaR}_\alpha$6, and vice versa (Barkoutsos et al., 2019). The paper’s two-qubit example makes the point starkly: for

$\mathrm{CVaR}_\alpha$7

the expectation objective is constant in $\mathrm{CVaR}_\alpha$8, so every parameter value is a global minimum and no optimization is possible, whereas at $\mathrm{CVaR}_\alpha$9 the CVaR objective becomes

$\lceil \alpha K\rceil$0

which has nontrivial local structure and drives the state toward increased ground-state sampling probability (Barkoutsos et al., 2019).

The global minima story is more nuanced. If the variational form can reach the ground state and $\lceil \alpha K\rceil$1 is diagonal, then a global optimum of the expectation objective also minimizes $\lceil \alpha K\rceil$2 for any $\lceil \alpha K\rceil$3. Conversely, states with overlap $\lceil \alpha K\rceil$4 on the ground state are global minima of $\lceil \alpha K\rceil$5 for $\lceil \alpha K\rceil$6, even if they are not minima of the expectation objective (Barkoutsos et al., 2019). This is one source of the method’s practical utility: the objective can already be minimized once the circuit has moved sufficient probability mass onto optimal samples, without requiring the full distribution to be sharply concentrated.

The same papers also emphasize two caveats that are often overlooked. First, small $\lceil \alpha K\rceil$7 produces a “soft cap”: once the overlap with the optimum exceeds $\lceil \alpha K\rceil$8, further increasing that overlap need not improve the CVaR objective (Barkoutsos et al., 2019). Second, CVaR does not remove ansatz expressivity bottlenecks. In the QAOA analysis of the original study, small-depth circuits can remain “flat,” keeping amplitudes exponentially small on individual bit strings; CVaR can still guide optimization by focusing on the low-energy tail, but shallow-depth reachability remains a bottleneck (Barkoutsos et al., 2019). Later experiments found the same qualitative pattern: Ascending-CVaR improved QAOA overlap relative to fixed CVaR, yet absolute overlaps remained low at shallow depth for Max-Cut, Number Partitioning, and Portfolio Optimization (Kolotouros et al., 2021).

4. Scheduled CVaR objectives and smooth surrogates

A major extension of the fixed-$\lceil \alpha K\rceil$9 formulation is the evolving-risk objective known as Ascending-CVaR. In that approach the objective at optimization stage $H$0 is

$H$1

with a risk level $H$2 that increases during training. The reported schedules include a linear update

$H$3

with $H$4 in 15–20 qubit VQE experiments, and a sigmoid schedule with $H$5 that spends more early iterations at small $H$6 and was beneficial for harder Number Partitioning instances (Kolotouros et al., 2021).

The theoretical justification is continuation across objective landscapes. The paper reports that if the ansatz can achieve maximum overlap $H$7 with the ground state, then for any $H$8, minimizing $H$9 preserves the same global optimum as minimizing the energy. It also reports that if $n$0 is a global minimizer for $n$1 with $n$2, then it is also a global minimizer for any $n$3 with $n$4, while local minima can change as $n$5 changes (Kolotouros et al., 2021). The intended effect is to use small-$n$6 phases to identify promising basins and larger-$n$7 phases to erase suboptimal minima without perturbing the true optimum.

Not all schedules widen the tail. In dynamic portfolio optimization, a descending adaptive schedule was introduced:

$n$8

with $n$9, $|z\rangle$0, and $|z\rangle$1, decreasing every 48 NFT parameter updates (Haghighi, 8 Jun 2026). There the purpose is explicitly different: start with a broad, stable signal to steer amplitude toward feasible or near-feasible regions, then tighten the tail to refine high-quality samples. The literature therefore contains both widening and tightening schedules. This suggests that schedule direction is task-dependent: Ascending-CVaR addresses spurious local minima in unconstrained or lightly constrained landscapes, whereas descending schedules can prioritize feasibility discovery before tail refinement.

A separate line of development places CVaR inside a smooth tilted-loss framework. The Quantum Tilted Loss is defined by

$|z\rangle$2

It interpolates continuously between expectation minimization and tail-sensitive objectives, preserves the true global minima, and admits exact parameter-shift gradients for suitable gate generators (Qiu et al., 4 May 2026). For $|z\rangle$3, the paper proves the entropic lower bound

$|z\rangle$4

and defines the corresponding EVaR surrogate by optimizing the right-hand side over $|z\rangle$5 (Qiu et al., 4 May 2026). The same work formalizes the trade-off that sharper tilting can improve trainability while increasing finite-shot variance, shifting the bottleneck from vanishing gradients to measurement sampling variance.

5. Applications and empirical record

The empirical record spans simulation, small-device demonstrations, utility-scale sampling workflows, and constrained optimization with nonlinear penalties. Across these settings, the common pattern is that tail-focused objectives improve the probability of sampling good bit strings, but the effective choice of $|z\rangle$6, optimizer, and shot budget remains problem-dependent.

The original CVaR-VQA study evaluated maximum stable set, Max3Sat, Number Partitioning, MaxCut, Market Split, and Portfolio Optimization across 340 random instances and 14,280 test cases. In simulation, CVaR-VQE improved rapidly as $|z\rangle$7 decreased and circuit depth increased; with VQE depth $|z\rangle$8 and $|z\rangle$9, almost all instances reached at least $z\in\{0,1\}^n$0 overlap within 50 normalized iterations, whereas at $z\in\{0,1\}^n$1 only about $z\in\{0,1\}^n$2 did so. On IBM Q Poughkeepsie, a 6-qubit portfolio optimization experiment with 8192 shots per evaluation found that $z\in\{0,1\}^n$3 and $z\in\{0,1\}^n$4 quickly increased ground-state overlap to the $z\in\{0,1\}^n$5-level in each of five runs, while $z\in\{0,1\}^n$6 remained very small and largely flat (Barkoutsos et al., 2019).

Ascending-CVaR extended this line in noiseless emulation for Max-Cut, Number Partitioning, and Portfolio Optimization using VQE and QAOA. In VQE, Ascending-CVaR reportedly outperformed both the expectation objective and constant CVaR on all tested cases, achieving up to ten times greater overlap in Portfolio Optimization and Number Partitioning and an $z\in\{0,1\}^n$7 average improvement in Max-Cut. In hard Number Partitioning instances, standard objectives failed in almost all cases, constant CVaR found the correct solution in about $z\in\{0,1\}^n$8 of cases, and Ascending-CVaR did so in about $z\in\{0,1\}^n$9 (Kolotouros et al., 2021).

Later work generalized the sampling-based pattern to constrained losses evaluated directly on bit strings. In multi-dimensional knapsack, a slack-free step-penalty formulation combined with CVaR at $|\psi(\theta)\rangle$00 and 4000 shots per evaluation outperformed slack-based QUBO on all 12 reported instances in mean optimality gap; for the largest reported instance, pet7 with $|\psi(\theta)\rangle$01, the median gap was about $|\psi(\theta)\rangle$02 for the finite-sampling mean objective and about $|\psi(\theta)\rangle$03 for CVaR (Lee et al., 22 Apr 2026). In ETF construction, a 109-qubit hardware experiment on IBM Heron processors, using $|\psi(\theta)\rangle$04 shots, $|\psi(\theta)\rangle$05, and local-search post-processing, reached a best relative solution error of $|\psi(\theta)\rangle$06 on a simplified 109-bond instance (Agliardi et al., 19 Aug 2025). In 150-qubit dynamic portfolio optimization, simulator studies selected an adaptive CVaR schedule, PSO$|\psi(\theta)\rangle$07NFT optimizer scheduling, and hardware-aware layouts; subsequent ibm_quebec experiments found that the heavy-hex-native deep-chain layout achieved the best final CVaR-tail value and best sampled objective among the tested layouts, although no quantum advantage over SCIP was observed (Haghighi, 8 Jun 2026).

Study	Setting	Reported outcome
(Barkoutsos et al., 2019)	340 random instances; VQE/QAOA; also 6-qubit IBM hardware	Lower $|\psi(\theta)\rangle$08 improved overlap and convergence; hardware runs at $|\psi(\theta)\rangle$09 outperformed $|\psi(\theta)\rangle$10
(Kolotouros et al., 2021)	15–20 qubits; Max-Cut, Number Partitioning, Portfolio; VQE and QAOA	Ascending-CVaR exceeded constant CVaR and expectation; hard Number Partitioning reached about $|\psi(\theta)\rangle$11 success
(Lee et al., 22 Apr 2026)	MDKP with slack-free custom penalty; 4000 shots	Custom penalty + CVaR beat slack-based QUBO on all 12 instances
(Agliardi et al., 19 Aug 2025)	109-qubit ETF construction on IBM Heron	Best post-processed relative gap $|\psi(\theta)\rangle$12
(Haghighi, 8 Jun 2026)	150-qubit dynamic portfolio optimization	Adaptive CVaR and hardware-native layouts improved practical performance; no quantum advantage over SCIP

6. Limitations, misconceptions, and evaluation methodology

A recurring misconception is that smaller $|\psi(\theta)\rangle$13 is unconditionally superior. The literature does not support that conclusion. Small $|\psi(\theta)\rangle$14 sharpens focus on elite samples, but it also raises estimator variance, introduces a soft cap on rewarded overlap, and can destabilize derivative-free optimization if shot counts are not increased accordingly (Barkoutsos et al., 2019). A later evaluation framework, applied to a 16-qubit QUBO using RealAmplitudes and COBYLA, found that $|\psi(\theta)\rangle$15 dominated lower $|\psi(\theta)\rangle$16 values in feasibility and quality at moderate-to-high shot counts, with diminishing gains beyond about 20,000 shots for most $|\psi(\theta)\rangle$17 (Mamedaliev et al., 21 Jan 2026). Another study likewise reported that fixed $|\psi(\theta)\rangle$18 without shot scaling performs poorly in dynamic portfolio optimization (Haghighi, 8 Jun 2026). This suggests that the effective risk level depends on the interaction among problem structure, ansatz, optimizer, and sampling budget.

A second misconception is that CVaR-VQA is defined only for Ising/QUBO expectation estimation. Several newer formulations use the quantum circuit purely as a sampler and compute a nonlinear classical loss on each observed bit string. This is how hinge-penalty ETF construction, step-penalty knapsack optimization, and dynamic multi-period portfolio objectives avoid explicit slack-variable Hamiltonians or costly exact expectation computation for nonlinear penalties (Agliardi et al., 19 Aug 2025). The trade-off is that qubit savings and modeling flexibility come at the price of heavier shot budgets and explicit classical post-processing.

Because sampling-based VQAs are stochastic at multiple levels, evaluation has itself become a research topic. A dedicated framework introduced three complementary metrics—feasibility, quality, and reproducibility—and a quality diagram that visualizes the trade-off between success probability and resource consumption. Reproducibility is quantified by Shannon entropy over binned outcome distributions, and algorithm selection is framed as a thresholded decision rule under resource constraints (Mamedaliev et al., 21 Jan 2026). This perspective treats CVaR-VQAs not as single trajectories but as distributions over performance outcomes.

The main technical limitations remain measurement complexity, non-smoothness, hardware noise, and ansatz reachability. The empirical CVaR objective depends on order statistics and can be discontinuous under finite sampling; exact parameter-shift does not apply directly to the hard-truncated estimator, which is one reason smooth surrogates such as QTL and EVaR have been proposed (Qiu et al., 4 May 2026). Device noise and readout error distort energy histograms and quantiles, making careful calibration important (Kolotouros et al., 2021). More fundamentally, if the ansatz cannot reach the ground-state subspace, CVaR preserves the true optimum as an objective target but cannot make it attainable (Qiu et al., 4 May 2026).

Sampling-based CVaR-VQA is therefore best understood not as a single algorithmic recipe but as a family of tail-sensitive hybrid optimization procedures. Its defining idea is stable across variants: use finite-shot samples from a parametrized quantum circuit, evaluate a classical optimization loss on those samples, and train on the lower tail rather than the full mean. The subsequent literature shows that this family encompasses fixed-risk CVaR, evolving risk schedules, smooth entropic surrogates, custom penalties for constrained optimization, and hardware-aware large-qubit workflows, with performance gains that are substantial in many reported settings but are tightly conditioned by shot allocation, optimizer design, and circuit expressivity (Barkoutsos et al., 2019).