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Take a four-node graph and split its nodes into two groups so as many connecting edges as possible cross between the groups. That is Max-Cut. A quantum approach does not simply list every possible split and choose the best one. Instead, it encodes each binary choice in a qubit, turns the objective into a cost Hamiltonian, uses interference to reshape the probabilities, and measures samples that favor low-cost (or high-value) assignments.
The practical method most often discussed for this task is the hybrid quantum-classical loop. Classical software builds the objective and updates parameters; a quantum processor prepares and samples a parameterized state. The result is a candidate generator and heuristic optimizer, not a universal replacement for classical optimization.
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Start with a graph: what the quantum computer is trying to optimize
Suppose four nodes—A, B, C and D—are connected by edges. Max-Cut asks for a two-way partition that maximizes the number of edges whose endpoints land in different groups. One possible answer is {A, C} and {B, D}. Every edge crossing between those sets contributes to the score; an edge whose endpoints stay in the same set contributes nothing.
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1Clear out junk files and repair common Windows errors2Fix the driver behind crashes, sound loss and screen glitches3Repair Windows errors before they cause bigger problemsWith n nodes, a binary variable can represent each node’s side of the partition: 0 for one group and 1 for the other. There are 2^n possible bit strings before accounting for the fact that swapping the labels of the two groups describes the same cut. Max-Cut is NP-hard, and the same pattern appears in scheduling, routing, clustering and resource-allocation tasks: many discrete choices interact, and a locally attractive choice can make the global result worse.
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IBM’s Quantum tutorial describes Max-Cut as a combinatorial optimization problem with applications in clustering, network science and statistical physics. It writes the binary objective as a quadratic unconstrained binary optimization (QUBO), commonly represented as minimizing xᵀQx. A transformation maps that binary expression to a quantum cost Hamiltonian. States with lower energy then correspond to better objective values (or, if the sign convention is reversed, higher-value solutions).
What a qubit contributes—and what it does not
Superposition is a compact representation
A classical bit is either 0 or 1. A qubit can be prepared in a state that combines both computational-basis states. For n qubits, the state can contain amplitudes for up to 2^n bit strings. NIST describes qubits as capable of being put into “superpositions of multiple states” and says entangled qubits can represent exponentially many combinations.
That representation is useful only if the algorithm manipulates amplitudes so that desirable strings become more likely to appear when measured. A measurement does not return the full list of amplitudes or every candidate assignment. It returns one classical bit string per shot, so an experiment must be repeated to estimate which strings occur often and what objective value they have.
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Optimization variables are rarely independent. In a graph, the value of assigning one node to a group depends on neighboring assignments. Entangling operations create correlations between qubits, allowing the state to describe relationships that a product of independent single-qubit states cannot capture efficiently.
Interference is the selection mechanism
Superposition alone does not favor a good cut. QAOA-style circuits apply phase changes based on the objective and then mix amplitudes between alternatives. Constructive interference can increase the probability of useful assignments, while destructive interference suppresses some poor ones. The circuit parameters determine how that reshaping occurs; finding effective parameters is part of the classical optimization loop.
“But contrary to popular belief, this doesn’t allow quantum computers to do an efficient ‘brute force’ search over all the potential solutions.”
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Stephen Jordan, Google quantum computing researcher and former NIST staff member, quoted by NIST
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How a QUBO becomes a quantum cost Hamiltonian
A QUBO uses binary variables xᵢ ∈ {0,1} and a quadratic objective such as xᵀQx. Linear terms score individual choices; quadratic terms score pairs, such as whether two connected nodes are placed on opposite sides of a cut. Constraints that are not naturally unconstrained can be folded into the objective with penalty terms. A penalty must be large enough to discourage an invalid assignment, but not so large that it overwhelms the real objective and makes parameter search difficult.
Quantum hardware commonly uses Pauli-Z operators whose eigenvalues represent binary choices. A substitution such as xᵢ = (1 − zᵢ)/2, with zᵢ represented by a Z operator, converts the QUBO into a Hamiltonian made from weighted products of Z operators. Because these terms are diagonal in the computational basis, measuring a bit string gives a directly evaluable energy. The lowest-energy measured strings are the strongest candidates under that encoding.
The encoding is a modeling decision, not an automatic translation of a business problem. Missing constraints, incorrect penalty weights or an objective that does not reflect operational costs can produce an excellent quantum answer to the wrong question.
QAOA in simple terms: the hybrid quantum-classical loop
IBM defines the Quantum Approximate Optimization Algorithm (QAOA) as “a hybrid quantum-classical iterative method for solving combinatorial optimization problems.” The quantum processor samples a parameterized circuit; a classical optimizer changes the parameters to improve the observed objective.
1. Prepare all binary assignments as a starting state
Initialize the qubits to |0⟩ and apply a Hadamard gate to each one, producing H^⊗n|0⟩. In an ideal noiseless circuit this creates an equal-amplitude starting superposition over the computational-basis strings. It is a broad starting point, not a completed search.
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2. Apply the cost operator
Use the Hamiltonian derived from the QUBO to apply phase rotations. A parameter usually called γ controls the strength or duration of this cost evolution. Assignments with different objective values acquire different phases.
3. Apply a mixer operator
A mixer, often built from X rotations, moves amplitude among computational-basis states. Its parameter is commonly written β. The alternating cost-and-mixer layers create the QAOA circuit. The number of repetitions is called the depth or layer count, often denoted p.
4. Measure many shots
Measure every qubit to obtain classical bit strings. Each shot is one candidate assignment. Compute its original objective value and check its constraints. A histogram shows how frequently each candidate appeared, while the best observed feasible assignment may be more useful than the most frequent one.
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The classical side aggregates objective values from the samples, estimates an expectation value or another score, and proposes new parameters. The circuit is run again with those parameters. This loop continues until the optimizer stops improving, reaches a budget, or encounters a noise- or hardware-limited plateau.
- Build the graph, QUBO and constraint penalties with classical code.
- Compile the cost and mixer circuits for the selected quantum backend.
- Choose initial
γandβvalues and a layer countp. - Execute the circuit for a selected number of shots.
- Decode bit strings, reject or penalize infeasible assignments, and calculate objective values.
- Use a classical optimization method to propose the next parameters.
- Repeat, then compare the best feasible sample with a strong classical baseline.
IBM’s tutorial includes a 100-node utility-scale QAOA example and notes that a small example can use a backend with 127 or more qubits. Those figures describe tutorial-scale demonstrations and hardware requirements for the stated examples, not a general rule that every 100-variable problem needs exactly 127 physical qubits.
Why these problems are called “complex”
Complexity is not just the number of variables. It also comes from constraints and interactions. A delivery plan may couple vehicle capacity, time windows, depot assignment and road choices. A resource-allocation model may couple machines, staff, budgets and deadlines. Mapping such a task to QUBO requires deciding which variables are binary, how many auxiliary variables are needed, and how every hard constraint becomes a penalty or a restricted mixer.
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QAOA is therefore most natural when a problem can be expressed as a QUBO or a related Hamiltonian without an explosion in variables, penalties or circuit terms. Route planning and resource allocation are plausible application areas, but an application needs a faithful objective, a valid constraint model and a useful decoding procedure before a quantum experiment can say anything about the real operation.
QAOA versus quantum annealing
Both approaches use quantum effects for optimization, but they are not the same algorithm or hardware model.
| Aspect | QAOA | Quantum annealing |
|---|---|---|
| Computational model | Gate-based, parameterized circuit with alternating cost and mixer layers | A different physical/computational approach that evolves a system toward low-energy configurations |
| Parameter control | Classical optimizer updates parameters such as γ and β between circuit evaluations | Uses an annealing schedule and hardware-specific controls rather than QAOA’s layer parameters |
| Output | Samples measured from a parameterized circuit | Samples from the final state of the annealing process |
| Typical formulation | QUBO or Hamiltonian implemented as gate operations | Optimization problems represented in an annealing-compatible energy landscape |
| Evidence status | Near-term work is heuristic; broad practical speedup is not established | NIST distinguishes annealing from the logic-gate model and discusses possible advantages for certain optimization problems cautiously |
Quantum annealing should not be described as simply “QAOA with more layers.” It has different physics, control methods and performance questions. In either case, the relevant test is whether the complete workflow improves a useful metric against the best available classical method.
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Noise and fragile states
Superposition and entanglement can be disrupted by stray fields, temperature fluctuations and cosmic rays. NIST describes the best systems in its 2026 page update as having hundreds of interconnected qubits and roughly one error in every thousand operations. An error rate at that scale can materially alter a deep optimization circuit, especially when many two-qubit gates are required.
Qubit count is only one resource
A model may need one logical qubit per binary variable plus auxiliary variables for constraints. Hardware connectivity can force additional routing gates. Circuit depth, two-qubit-gate count, calibration drift, measurement error and the number of shots can matter as much as the headline qubit total.
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Error correction is not yet a near-term assumption
IBM notes that useful near-term experiments cannot require enormous qubit counts or circuit depths beyond demonstrated literature, and that many error-corrected algorithms are not yet feasible. A small, shallow demonstration can run on present devices while a production-scale, fault-tolerant version of the same mathematical problem remains out of reach.
Can quantum computers beat classical optimization?
There is no established universal answer. NIST reports that scientists have published claims of quantum advantage, while also noting that early demonstrations have not proved truly useful and that classical computers have sometimes matched or exceeded quantum processors. IBM Research treats optimization advantage as an active research objective involving algorithm development and rigorous benchmarking, not as a settled result.
Near-term QAOA is best viewed as a heuristic search for a practical “sweet spot.” It may produce good solutions for a selected instance, but an advantage claim must account for the entire workflow rather than only the time spent on the quantum chip.
Use these comparison axes
| Metric | Question to ask |
|---|---|
| Objective quality | How close is the best or average feasible solution to the known optimum or strongest classical result? |
| Sampling probability | How often does the circuit produce a high-quality feasible assignment? |
| Total runtime | Does the measurement include queue time, circuit compilation, repeated shots and classical parameter updates? |
| Hardware resources | How many physical and logical qubits, layers and two-qubit gates are required? |
| Noise handling | What error mitigation, calibration procedure or post-processing is used, and what overhead does it add? |
| Classical baseline | Was the comparison made against a competitive solver configured for the same instance and stopping rule? |
IBM Research’s benchmarking program emphasizes rigorous quantum/classical metrics because a faster quantum subroutine can still lose overall if data preparation, queueing, sampling or classical optimization dominates.
A practical decision framework for an optimization team
- Specify the business objective. Write the exact quantity to minimize or maximize, including units, priorities and acceptable trade-offs.
- Separate hard constraints from preferences. Decide which violations are impossible and which can be penalized.
- Build a classical reference. Record solution quality and end-to-end runtime using the strongest practical solver available.
- Formulate a small QUBO instance. Verify the encoding by exhaustively checking tiny cases where every assignment can be enumerated classically.
- Estimate hardware cost. Count variables, auxiliary bits, Hamiltonian terms, circuit depth, connectivity overhead and required shots.
- Run noise-aware experiments. Compare ideal simulation, noisy simulation and hardware results when possible, keeping the same objective and decoding rules.
- Report distributions, not one lucky shot. Show objective-value distributions, feasibility rates and confidence intervals or repeated-run variation.
- Scale only after validation. A correct small encoding and a reproducible baseline matter more than an impressive qubit count.
What measurement really tells you
After the final circuit, measurement produces samples such as 010110.... The sample is a complete candidate assignment for the encoded variables, but the experiment does not reveal the amplitudes of all other assignments in that run. Repeating the circuit estimates a probability distribution. A high probability for a good state is useful; a single occurrence of a good state may not be, especially if millions of shots or extensive post-selection were required.
Always decode the measured string back into the original variables, check every constraint, and calculate the original objective rather than relying only on the Hamiltonian energy. Penalty terms can make an invalid low-energy state appear attractive if their weights were chosen poorly.
Where the field stands
NIST physicist Scott Glancy characterized the field this way: “It seems to me we’re just on the threshold of quantum systems doing genuinely new simulations that we can’t do classically.” That optimism does not remove the present optimization caveats. Current processors are noisy, measurement is limited, circuit depth is constrained, and the strongest classical methods continue to improve.
For optimization, the defensible claim today is narrower: qubits provide a new way to represent and sample correlated assignments, and algorithms such as QAOA provide a structured hybrid procedure for exploring them. Whether that procedure beats classical optimization depends on the instance, encoding, hardware, noise, parameter-search cost and baseline. Treating “quantum” as a synonym for “tries every answer at once” obscures the engineering work that determines the result.
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