Solving the Optimal Trading Trajectory Problem Using a Quantum Annealer

• The manager chooses investments across N assets and T time steps using return and risk forecasts while accounting for permanent and temporary market impact costs.Source: A. Introduction · p. 1
• A trajectory that jointly optimizes risk-adjusted returns can outperform the sequence of portfolios optimized independently at each time step.Rebalancing between statically optimal portfolios incurs costs.Source: A. Introduction · p. 1
• Trading in large or fixed lots makes the portfolio problem inherently discrete, so continuous solutions become approximations.The passage cites institutional even-lot constraints and large-block trading as examples.Source: A. Introduction · pp. 1–2
• The discrete problem is non-convex and remains difficult when the covariance matrix is ill-conditioned or degenerate.Source: A. Introduction · p. 2

• Single-period discrete portfolio optimization is NP-complete, with prior work using branch-and-bound and heuristic algorithms.Reported classical studies include problems involving up to 200 assets.Source: B. Previous work · p. 2
• Continuous multi-period portfolio optimization has been solved analytically through dynamic programming when the covariance matrix is positive definite.Source: B. Previous work · p. 2
• Analytical trade-execution work motivates including both temporary and permanent price-impact terms.Source: B. Previous work · p. 2
• The paper investigates implementing a discrete multi-period portfolio problem on D-Wave’s quantum annealer, following a proposal that such problems are suitable for quantum annealing.Source: B. Previous work · p. 2

• The formulation maximizes returns while incorporating risk, temporary market impact, permanent market impact, and other transaction costs.Source: C. Integer formulation · pp. 2–3
• Holdings must sum to K at every time step.Source: C. Integer formulation · p. 2
• Each asset’s holdings are constrained by a maximum allowed level K′.Source: C. Integer formulation · p. 2
• The objective’s risk term uses the forecast covariance tensor Σ and risk-aversion parameter γ, while its return term is forecast returns µ multiplied by holdings w.Source: C. Integer formulation · p. 2
• Transaction-cost terms penalize changes in holdings and capture temporary costs, direct costs, and permanent price impact.Source: C. Integer formulation · p. 3

• Optimal trade execution can be represented by changing the holdings constraint and extending transaction-cost and permanent-impact terms through time T + 1 with wT +1 = 0.Source: D. Extensions · p. 3
• Trajectory variance provides an alternative risk measure when covariance estimates are unreliable because of limited data or illiquidity.Source: D. Extensions · p. 3
• The variance-based formulation avoids covariance-matrix estimation and does not assume a normal return distribution.Source: D. Extensions · p. 3
• Because variance is quadratic in the returns, it can be included in a quadratic integer formulation.Source: D. Extensions · p. 3

Quantum annealing

• Quantum annealing continuously transforms an easy ground-state problem into the target Hamiltonian, with repeated runs increasing the probability of finding the optimum.Because real devices experience external interference, quantum annealers function as heuristic solvers.Source: A. Quantum annealing · p. 3
• Quantum tunnelling has been proposed to improve solution-space search and possibly speed optimization for certain problem classes.Source: A. Quantum annealing · p. 3

D-Wave hardware

• D-Wave’s annealer minimizes unconstrained binary quadratic functions, using a cryogenic, shielded, high-vacuum environment to reduce disturbances.Source: A. Quantum annealing · pp. 3–4

Quantum speedup

• Whether D-Wave provides quantum speedup remains an open question, despite a reported 108 speedup over simulated annealing on a specially constructed problem class.Source: A. Quantum annealing · p. 4

D-Wave hardware

• Chimera hardware contains bipartite eight-qubit unit cells connected in a square lattice, with q = 8s^2 physical qubits.Source: A. Quantum annealing · p. 4

Embedding limits

• For dense square Chimera embeddings, the largest problem has V = 4s + 1 logical variables; a 1100-qubit annealer embedded approximately Vb ≃ 140 variables at density ≃ 0.1.Source: A. Quantum annealing · p. 4

Encoding schemes

• Four encodings—binary, unary, sequential, and partitioning—recast integer holdings into binary variables for quantum annealing.Source: B. From integer to binary · pp. 4–5

Encoding selection

• The preferred encoding depends on the optimization problem and annealer, and Table III compares variable counts across example multi-period portfolio problems.Source: B. From integer to binary · p. 5, Table II · p. 6

Encoding trade-offs

• Binary, unary, and sequential encodings trade binary-variable efficiency against the largest representable integer because efficient encodings use larger noise-sensitive coefficients.Source: B. From integer to binary · p. 5
• Binary encoding uses the fewest variables but represents the smallest integer range, whereas unary uses more variables but represents the largest range.Source: B. From integer to binary · p. 5

Partition encoding

• Partition encoding requires exponentially many variables but can impose complicated constraints by excluding invalid partitions, while embedding may require substantially more physical qubits.Source: B. From integer to binary · p. 5

Encoding trade-offs

• Binary and sequential encodings may require penalties for infeasible holdings when the maximum holding is not exactly representable at the chosen bit depth.Source: B. From integer to binary · p. 5, Table I · p. 5

Constraint penalties

• Constraints are incorporated by rearranging, squaring, and summing constraint equations into a penalty term weighted by M > 0.Source: B. From integer to binary · p. 5
• An overly large penalty strength M increases sensitivity to system noise, so the paper selects M empirically by trial and error.Source: B. From integer to binary · p. 5

Experimental setup

• 200 random instances per problem were tested with one quantum-annealer query using 1000 reads, on hardware with 512 or 1152 qubits.The number of active qubits was slightly smaller than the nominal hardware count.Source: C. Numerical Results · p. 6

Read-count effects

• Success rates rise with more reads for fixed problem size, while fixed read counts are expected to reduce success as problem size increases.This read-count dependence helps explain the reported results across problem sizes.Source: C. Numerical Results · p. 6

Evaluation metric

• Success S(α) measures the percentage of instances whose annealer result falls within α% of optimality under Gaussian perturbations of the problem matrix.Each instance was perturbed at least 100 times and compared with exhaustive-solver optima.Source: C. Numerical Results · pp. 6–7

Limitation

• The variance of the success rate could not be statistically assessed because the required experimental runs exceeded available machine time.This limits uncertainty characterization of the reported success rates.Source: C. Numerical Results · p. 7

Software enhancements

• Custom software and parameter tuning produced a large improvement across all problems on the same 512-qubit hardware.The improvements combined an embedding solver, tuned identification coupling strengths, and multiple random gauges.Source: C. Numerical Results · p. 8, Table V · p. 8

Hardware limitations

• Intrinsic noise and coefficient-range scaling can reduce success rates, especially for this high-precision portfolio problem.The current chip’s intrinsic noise is estimated at 2%–4% of full scale.Source: D. Discussion · p. 9
• Sparse hardware connectivity requires identifying multiple physical qubits with one binary variable, adding strong couplings and potentially further scaling.This can reduce additional coefficients below ϵ and impact success rates.Source: D. Discussion · p. 9
• More qubits could enable larger problems and less noise-sensitive encodings, while inactive qubits and couplers reduce effective hardware density.Unary encoding is given as an example of an encoding less sensitive to noise than binary encoding.Source: D. Discussion · p. 9

Potential improvements

• Improved embeddings, calibration, majority voting, and random gauge selection are software or operational enhancements that may improve solutions.The pi-elite metric was used to select scaling, qubit-chain strength, and embeddings; random gauges were combined because success rates vary strongly across gauges.Source: D. Discussion · pp. 9–10

Runtime and speedup

• For the small-scale problems studied, time to solution is comparable on classical hardware and the quantum annealer.Quantum speedup for the optimal trading trajectory problem is expected only after speedup is demonstrated for general high-precision coupling problems.Source: D. Discussion · p. 10

III. CONCLUSIONS

• D-Wave’s quantum annealer demonstrated potential for high success rates on an important, difficult multi-period portfolio optimization problem.The authors characterize this as a limited experiment.Source: §III · p. 10
• Fine-tuning the quantum annealer produced a considerable improvement in success rates.The conclusion attributes this improvement to fine-tuning the annealer’s operation.Source: §III · p. 10
• Current quantum annealers solve only small problems, but future technological improvements are expected to support larger problems at higher success rates.The paper notes that larger problems are expected to be intractable on classical computers.Source: §III · p. 10

APPENDIX A DEFINITION OF SYMBOLS

• The notation defines w_t and µ_t as the t-th columns of their respective matrices, Σ_t as the t-th covariance-matrix page, and Δw_t as the change in holdings.Temporary and permanent transaction costs are represented by tensors Λ and Λ′, respectively.Source: Appendix A · p. 10