Problems and constructions¶

Every fit is described by the problem (which \(f\)) and the spectrum it acts on: eigenvalues or singular values, and where they come from. A third choice, the coefficient-selection method, is covered on the Method page.

Spectral information¶

A spectral interval is required. Give it either as bounds or as a matrix:

solve_fe_coefficients("inversion", sigma_min=0.5, sigma_max=3.0)
solve_fe_coefficients("inversion", matrix=A)

Passing neither, or both, raises an error. With matrix=, felcu computes eigenvalues or singular values itself.

spectral_kind says which data the bounds (or the matrix) give:

Problem	`spectral_kind='auto'` (default)	Also available
`identity` (or `action`)	eigenvalues; singular values for a non-Hermitian matrix	the other one
`inversion`	eigenvalues; singular values for a non-Hermitian matrix	the other one
`exponentiation`	eigenvalues	none
`square_root`, `logarithm`	eigenvalues (must be positive definite)	none

Eigenvalues need a Hermitian matrix (tested to hermitian_tol). Singular values are only available for the odd functions, identity and inversion (see Singular values).

The scaled variable¶

Following Sec. II of the paper, the fit is carried out in a scaled variable \(x\) on the fitted interval \(\Omega_\eta = [-\pi/\eta, \pi/\eta]\), strictly inside one period \(\mathbb T = [-\pi, \pi)\). This is what makes it a Fourier extension: the series \(g_m\) only has to match the target on part of the period, so a smooth non-periodic function can still be fitted to high accuracy.

Every fit is centred on the spectrum. With \(\mu\) and \(\delta\) the midpoint and half-width of the fitted spectrum, the scale \(\tau = \pi/(\eta\delta)\) maps it onto \(\Omega_\eta\), giving the scaled matrix \(\mathcal G = \tau(\mathcal H - \mu I)\) and the fitted function

\[ g(x) = f\!\left(\mu + \frac{x}{\tau}\right), \qquad f(\mathcal H) = g(\mathcal G). \]

result.mu and result.tau hold \(\mu\) and \(\tau\). The block encoding then implements \(g_m(\mathcal G) = \sum_k c_k e^{ik\mathcal G}\), where each \(e^{ik\mathcal G} = e^{-ik\tau\mu} e^{ik\tau\mathcal H}\) is a Hamiltonian simulation of \(\mathcal H\) up to a known phase.

Centring means the series only covers the half-width: for an interval far from zero, such as \([2, 5]\), it fits \([-1.5, 1.5]\) rather than \([-5, 5]\), and the constant part \(c_0 I\) carries \(f(\mu)\).

Problem	Spectrum	Fitted \(g(x)\)	Basis	Fitted set
identity	eigenvalues	\(\mu + x/\tau\)	constant + sine	\(\Omega_\eta\)
identity	singular values	\(x/\tau\)	odd sine	\(\Omega_\eta\)
inversion	definite eigenvalues	\((\mu + x/\tau)^{-1}\)	full	\(\Omega_\eta\)
inversion	indefinite eigenvalues	\((\mu + x/\tau)^{-1}\)	full	\(\Omega_\eta\) less the gap
inversion	singular values	\(\tau/x\)	odd sine	\(\Omega_\eta\) less the gap
exponentiation	eigenvalues	\(e^{\mu + x/\tau}\)	full	\(\Omega_\eta\)
square root	positive eigenvalues	\(\sqrt{\mu + x/\tau}\)	full	\(\Omega_\eta\)
logarithm	positive eigenvalues	\(\log(\mu + x/\tau)\)	full	\(\Omega_\eta\)

Bases¶

odd sine: \(a_0 = 0\) and \(b_k = 0\); only sine terms. Used for singular values, which need an odd series.
constant + sine: a constant plus sine terms.
full: constant, sine and cosine terms.

Restricting the basis halves the number of unknowns and guarantees the right symmetry exactly.

Singular values¶

A general matrix \(A\) has no real eigenvalue interval, so it is handled through its Hermitian dilation

\[ \mathcal H = \begin{pmatrix} 0 & A \\ A^\dagger & 0 \end{pmatrix}, \qquad \operatorname{spec}(\mathcal H) = \pm\operatorname{sv}(A). \]

The dilation's spectrum is symmetric, so its midpoint is \(\mu = 0\) and its half-width is \(\Sigma_{\max}\): the same centred construction, with \(\tau = \pi/(\eta\Sigma_{\max})\). With \(A = U\Sigma V^\dagger\), an odd \(g\) gives

\[ g(\mathcal H) = \begin{pmatrix} 0 & U g(\Sigma) V^\dagger \\ V g(\Sigma) U^\dagger & 0 \end{pmatrix}, \]

so the series is restricted to odd sine terms. For inversion the lower-left block is \(A^{-1}\).

Inversion¶

\(x^{-1}\) has a pole at zero. On a definite interval (both bounds the same sign, neither zero; negative-definite matrices work too), the pole lies outside the spectrum, and the default \(\eta\) puts it on the periodic boundary.

On an indefinite interval the spectrum lies on both sides of the pole, \(\operatorname{spec}(\mathcal H) \subseteq [\lambda_{\min}, a] \cup [b, \lambda_{\max}]\) with \(a < 0 < b\). The fit covers both parts and leaves the gap around the pole unconstrained:

\[ \Omega = [-x_{\max}, \tau(a - \mu)] \cup [\tau(b - \mu), x_{\max}], \qquad x_{\max} = \frac{\pi}{\eta}. \]

result.x_gap holds the gap in the scaled variable. Give the gap with gap=(a, b); with matrix=, it is read from the eigenvalues either side of zero, and a matrix with an eigenvalue at zero is rejected as singular. When the gap reaches an end of the spectrum, as for a matrix with a single negative eigenvalue, that side of \(\Omega\) is a single point. It is still fitted, weighted as much as the whole set, since as a measure it would carry no weight:

solve_fe_coefficients("inversion", sigma_min=-2.0, sigma_max=3.0, gap=(-0.5, 0.4))
solve_fe_coefficients("inversion", matrix=H)  # indefinite Hermitian H

Singular-value inversion is the symmetric case, \(a = -\Sigma_{\min}\) and \(b = \Sigma_{\min}\), which is the disconnected set \(\Omega^{\rm indef}_{\eta,\kappa}\) of Sec. VI C of the paper. A general indefinite interval is not centred on the pole, so \(\mu \neq 0\).

Square root¶

\(\sqrt{x}\) has a branch point at zero and is only defined (as a real function) for \(x > 0\). It follows the paper's treatment of the matrix square root (Sec. VI D): \(\mathcal H\) must be positive definite, \(0 < \lambda_{\min} \le \lambda_{\max}\), and the default \(\eta\) places the branch point on the periodic boundary rather than near the fitted endpoint.

\(\sqrt x\) is concave, so it does not satisfy the convex endpoint-reflection condition and the subnormalisation is bounded but not optimal: the paper shows \(\alpha_m = O(\kappa^{(2w-1)/(4w)}\sqrt{\lambda_{\max}})\), an extra \(O(\kappa^{1/4})\) for \(w = 1\).

Logarithm¶

\(\log x\) also has a branch point at zero and is treated like the square root: \(\mathcal H\) must be positive definite, and the default \(\eta\) places the branch point on the periodic boundary. All three methods support it. The default Sobolev scale is

\[ r_{\log,w} = \kappa^{-(2w-1)/(2w)} (1 + \ln\kappa)^{-1/w}, \qquad \kappa = \lambda_{\max}/\lambda_{\min}, \]

the square-root rule with an extra factor for the logarithm. When the spectrum straddles \(1\), \(\log\mathcal H\) has eigenvalues of both signs and \(\|\log\mathcal H\| = \max(|\log\lambda_{\min}|, |\log\lambda_{\max}|)\). \(\log x\) is concave, but the sign-adjusted profile \(-\log x\) is convex and decreasing, so it is largest at \(\lambda_{\min}\). Flipping the sign of \(f\) only flips the sign of every coefficient. With endpoint reflection at \(\eta = 2\), convexity phase-aligns every non-constant Fourier mode at \(\lambda_{\min}\), and the constant term \(c_0\) is the mean of \(-\log x\) over the fitted interval. So the subnormalisation is optimal, \(\alpha_m = \lvert\log\lambda_{\min}\rvert - \epsilon_m\), whenever that mean is non-negative:

\[ \frac{1}{\lambda_{\max} - \lambda_{\min}} \int_{\lambda_{\min}}^{\lambda_{\max}} (-\log x)\,dx = 1 - \frac{\lambda_{\max}\log\lambda_{\max} - \lambda_{\min}\log\lambda_{\min}}{\lambda_{\max} - \lambda_{\min}} \ge 0 . \]

This always holds when \(\lambda_{\max} \le 1\), where \(-\log x \ge 0\) throughout, but it can also hold when \(\lambda_{\max} > 1\), provided \(\lambda_{\min}\) is small enough: \([0.1, 1.5]\) and \([0.2, 2]\) both qualify. When the mean is negative, as on \([0.5, 2]\), \(\alpha_m\) is bounded but not optimal, as for the square root.

The extension parameter \(\eta\)¶

\(\eta > 1\) sets how much of the period is left free. Larger \(\eta\) gives the series more room to turn around, but packs the spectrum into a smaller interval. For method="sobolev" the defaults are:

Problem	Default \(\eta\)	Reason
identity, exponentiation	\(2\)	entire \(f\); optimal for reflected extensions
inversion (definite), square root, logarithm	\(\eta_{\rm def} = \lvert\mu\rvert/\delta = (\kappa + 1)/(\kappa - 1)\)	puts the pole or branch point \(x = -\tau\mu\) at the periodic boundary
inversion (indefinite or singular values)	\(1 + (b - a)/(2\delta)\)	the gap around the pole and the free part of the period are equally wide

For singular values (or any symmetric gap) the last rule is the paper's \(\eta_{\rm indef} = 1 + 1/\kappa\). Here \(\kappa = \max|\lambda|/\min|\lambda|\) (\(\eta = 2\) if \(\kappa = 1\)). Override with eta=, or query the default with default_eta, which also takes method=.

The explicit methods use their own defaults and ranges (see Method):

`method`	Default \(\eta\)	Allowed
`"reflected"`	\(2\)	\(\eta \ge 2\)
`"arcsine_taylor"`, identity and exponentiation	\(3\)	\(\eta > 2\)
`"arcsine_taylor"`, inversion, square root and logarithm	\(\min\{3,\ 1 + \lvert\mu\rvert/\delta\}\)	\(2 < \eta < 2\lvert\mu\rvert/\delta\)

Both explicit methods require \(\eta < 2|\mu|/\delta\) for inversion, square root and logarithm, and neither supports the two-interval fit of indefinite or singular-value inversion.

Problem name aliases¶

Canonical	Also accepted
`identity`	`action`, `matrix-action`
`inversion`	`inverse`
`exponentiation`	`exp`, `exponential`
`square_root`	`sqrt`, `square-root`
`logarithm`	`log`, `ln`

Names are case-insensitive. identity and action are interchangeable names for \(f(\mathcal H) = \mathcal H\); results always record the canonical name, result.problem == "identity".