On the detailed structure of quantum control landscape for fast single qubit phase-shift gate generation

TL;DR

The paper analyzes the detailed structure of the quantum control landscape for fast single-qubit phase-shhift gate generation by performing a spectral analysis of the Hessian at the special saddle point $f_0=0$. It reduces the Hessian to an injective integral operator and, via an ODE-based eigenproblem, derives explicit conditions and domain-based counts for positive and negative eigenvalues, providing magnitude estimates as well. The main result classifies the Hessian spectrum across the parameter domains $\mathcal{D}_1$–$\mathcal{D}_4$, showing purely negative spectra in $\mathcal{D}_1$, two positive directions in $\mathcal{D}_2$, and sign patterns depending on $\varphi_W+T$ in $\mathcal{D}_3$ and $\mathcal{D}_4$, thereby quantifying the optimization landscape near the saddle. These insights sharpen understanding of optimization difficulty near the saddle and support the broader absence-of-traps picture for fast-time quantum control, with magnitude estimates aiding practical convergence analyses.

Abstract

In this work, we study the detailed structure of quantum control landscape for the problem of single-qubit phase shift gate generation on the fast time scale. In previous works, the absence of traps for this problem was proven on various time scales. A special critical point which was known to exist in quantum control landscapes was shown to be either a saddle or a global extremum, depending on the parameters of the control system. However, in the case of saddle the numbers of negative and positive eigenvalues of Hessian at this point and their magnitudes have not been studied. At the same time, these numbers and magnitudes determine the relative ease or difficulty for practical optimization in a vicinity of the critical point. In this work, we compute the numbers of negative and positive eigenvalues of Hessian at this saddle point and moreover, give estimates on magnitude of these eigenvalues. We also significantly simplify our previous proof of the theorem about this saddle point of the Hessian [Theorem~3 in B.O.~Volkov, O.V.~Morzhin, A.N.~Pechen, J.~Phys.~A: Math. Theor. {\bf 54}, 215303 (2021)].

Figures (2)

• Figure 1: Illustration for Lemma \ref{['lemma2']} (left) and for Lemma \ref{['lemma3']} (right). Left: Two inclined straight lines have angles of inclination $T$ and $\tan(T)$, i.e., they are defined by equations $y=Tx$ and $y=\tan(T)x$, respectively. The line $y=\tan(Tx)$ on the interval $(0,1)$ lies between these two strait lines. Hence on the interval $(0,1)$ line $y=\alpha x$ intersects line $y=\tan(Tx)$ if and only if $T<\alpha<\tan(T)$. Right: The inclined straight line $y=\cot(T)x$ intersects with line $y=\cot(Tx)$ at $x=1$. Hence equation $\alpha x=\cot(Tx)$ on $(0,1)$ has one solution if $\alpha> \cot{T}$ and has no solutions if $\alpha\le\cot{T}$.
• Figure 2: The domains of the rectangle $(\varphi_W,T)\in[0,\pi]\times [0,\pi/2]$. $\mathcal{D}_3$ is the left square except of the dashed diagonal and borders. $\mathcal{D}_1$ is the bottom triangle in the right square with left vertical border and without the solid diagonal and bottom horizontal border. $\mathcal{D}_4$ is the vertical line marked with crosses. $\mathcal{D}"_2$ is the dash-dotted line. $\mathcal{D}_2'$ is the area between $\mathcal{D}_1$ and $\mathcal{D}_2"$. $\mathcal{D}_2"'$ is the area between $\mathcal{D}_2"$ and $\mathcal{D}_4$. Note that these notations for the domains $\mathcal{D}_2,\mathcal{D}_3,\mathcal{D}_4$ are different from that used in Volkov_Morzhin_Pechen. In the domain $\mathcal{D}_1$, the Hessian at critical point $f_0=0$ has only negative eigenvalues. On the dashed diagonal (in the left square), the critical point $f_0=0$ is a point of global minimum. On the solid diagonal (in the right square), the critical point $f_0=0$ is a point of global maximum. On the domain $\mathcal{D}_2=\mathcal{D}_2'\cup \mathcal{D}_2"\cup \mathcal{D}_2"'$, the critical point $f_0=0$ is a saddle point with two positive and infinitely many negative eigenvalues of Hessian. On the bottom triangle of $\mathcal{D}_3$, the critical point $f_0=0$ is a saddle point with one positive and infinitely many negatives eigenvalues of the Hessian. On the top triangle of $\mathcal{D}_3$ and on the domain $\mathcal{D}_4$, the critical point $f_0=0$ is a saddle point with one negative and infinitely many positive eigenvalues of the Hessian.

Theorems & Definitions (11)

• Definition 1
• Theorem 1
• Theorem 2
• Remark 1
• Theorem 3
• Theorem 4
• Lemma 1
• Lemma 2
• Proposition 1
• Proposition 2