QCPOP — Quantum Control via POlynomial oPtimization — looks for the global optimum of a quantum control problem instead of a local one. It approximates the control problem by a polynomial optimization problem, then uses methods developed in the last two decades that find, and can certify, the global optimum of polynomials. The method was introduced by Bondar, Balada Gaggioli, Korpas, Mareček, Vala and Jacobs [1], and extended to gate synthesis by Balada Gaggioli et al. [2].

Why global optimization

Gradient methods such as GRAPE, Krotov or GOAT converge to the nearest optimum. For unconstrained problems the control landscape is often free of traps, but realistic constraints — short durations, few control parameters, bounded amplitudes — can create many local optima. Simple examples have traps by construction [3]. Local methods must then be restarted from many initial guesses, and if the traps are dense enough they may never find a good solution. A global method removes this problem: one run returns the best solution of the (approximated) problem.

From quantum control to polynomials

Take a Hamiltonian \(H = H_0 + E(t)\,V\) and a figure of merit such as the Hilbert–Schmidt distance \(F = \lVert U(T) - U^\star \rVert^2\) or the infidelity \(1 - \big|\mathrm{Tr}\big(U(T)^\dagger U^\star\big)\big|/N\). QCPOP turns \(F\) into a polynomial in finitely many real variables with three approximations [1]:

  1. Magnus expansion. The propagator is written as \(U(T) = \exp(\Lambda_n)\) with \(\Lambda_n = \Omega_1 + \dots + \Omega_n\), where \(\Omega_1 = \int_0^T A(t)\,dt\), \(\Omega_2 = \tfrac12 \int_0^T\!\!\int_0^t [A(t), A(s)]\,ds\,dt\), … and \(A = -iH\) [4]. Every truncation is exactly unitary, and the series converges when \(\int_0^T \lVert A(t) \rVert\, dt < \pi\).
  2. Chebyshev expansion of the exponential. The matrix exponential is replaced by a truncated Chebyshev series with Bessel-function coefficients, a polynomial in the generator \(\Lambda_n\).
  3. A finite control basis. The control is expanded as \(E(t) = \sum_{k=0}^{m-1} x_k f_k(t)\), for example a power series \(f_k(t) = t^k\).

Substituting (3) into (1), each Magnus term becomes a polynomial in the coefficients \(x_k\); with (2), so does \(U(T)\), and therefore so does \(F\). A QCPOP model is labelled by the truncation orders \((n, p, m)\) — Magnus terms, Chebyshev terms and control parameters. In the examples studied, \(n = 3\) and \(p = 5\) were found to be sufficient.

For piecewise-constant controls the formulation is even simpler. With \(m\) segments of constant amplitude \(E_j\), \(U(T) = \prod_j \exp[-i(H_0 + E_j V)\,t_j]\), and only the Chebyshev expansion of each factor is needed — no Magnus expansion.

Solving the polynomial problem globally

A polynomial optimization problem asks for the global minimum of a polynomial \(f(\mathbf x)\), possibly subject to polynomial constraints \(g_i(\mathbf x) \ge 0\). Two families of methods are used:

  • The moment–sum-of-squares (SOS) hierarchy. Lasserre’s moment relaxations, and their dual SOS formulation, form a sequence of semidefinite programs whose optimal values are increasing lower bounds that converge to the global minimum under mild conditions [5, 6]. When a relaxation is tight, this is certified and the minimizer can be extracted. Exploiting sparsity (as in TSSOS) is essential to make the relaxations tractable [7].
  • Homotopy continuation. This method numerically tracks all solutions of the polynomial system of critical-point equations, so every extremum is found and the best one chosen [8].

In the original study, homotopy continuation worked better for the infidelity, and the moment–SOS hierarchy for the Hilbert–Schmidt distance [1].

What it has shown

  • Traps are no obstacle. On the three-level problem of Pechen and Tannor, whose objective has a trap at zero control [3], homotopy continuation applied to the QCPOP polynomial returns all extrema. A gradient method started near zero finds only the local one.
  • Gate synthesis. For 1000 random target unitaries of a three-level system, QCPOP reached better fidelities than GRAPE and CRAB, at a comparable time per run [1].
  • Dense traps. In a state-transfer problem between two coupled oscillators at very short times, gradient searches fail and need thousands of restarts. QCPOP found the best solution with three control segments in a single run, and showed that no two-segment solution exists [1].
  • Certified gates. Optimizing over the Hermitian generators of the unitaries instead of the unitaries themselves keeps the polynomials small. This gives single-shot gate synthesis with certificates of global optimality [2].

The same idea applies beyond control. Hamiltonian identification can be written as a polynomial optimization [1], and so can learning the most accurate Markovian (Lindblad) model of an open system from time-series data [9] — see System identification for control.

Computability

Deciding whether a given quantum control task is possible is uncomputable on a Turing machine: digitized quantum control can encode Diophantine equations, which no algorithm can solve in general [10]. Casting control as polynomial optimization gives a different picture in the Blum–Shub–Smale model of computation, which works with real numbers. There, quantum optimal control problems become computable, with a run time governed by the convergence of the Magnus expansion and of the moment–SOS hierarchy [1].

Strengths and limitations

Strengths

  • Finds the global optimum of the approximated problem in a single run, with certificates in the moment–SOS case.
  • Unaffected by traps, however dense — and can prove that no better solution exists, which reveals the true limits of a given control setup (Quantum speed limits).
  • The same machinery serves control, Hamiltonian identification and system identification.

Limitations

  • The size of the semidefinite programs grows quickly with the number of variables and the polynomial degree, so the number of control parameters is currently small; sparsity is crucial.
  • The optimum is that of the truncated problem, so the solution should be checked against the exact dynamics. The Magnus variant also requires \(\int \lVert A \rVert\, dt < \pi\).
  • A young method: tools and practical guidelines are still being developed.