Finite adiabatic path model

Let $V$ be a nonzero finite-dimensional complex inner-product space with inner product linear in its first argument. A finite adiabatic path is a family $H(s):V\to V$ of linear endomorphisms (Linear map between vector spaces over the same field) indexed by $s\in [0,1]$, whose matrix entries are twice continuously differentiable in $s$ and such that $H(s)=H(s{)}^{\ast }$ for every $s\in [0,1]$. A coherently followed spectral band is a norm-continuous family of orthogonal projections $P(s):V\to V$ indexed by $s\in [0,1]$, of one fixed rank $m\ge 1$, commuting with $H(s)$, whose range is a sum of full eigenspaces and which locally continues the same isolated cluster. For a finite union of positively oriented circles $\mathrm{\Gamma }$, define ${\oint }_{\mathrm{\Gamma }}F(z)dz$ as the sum of the parameterized integrals ${\int }_0^{2\pi }F(c+r{e}^{it})ir{e}^{it}dt$ over its circle components. Local continuation means that on each neighborhood there is such a contour enclosing exactly the selected cluster and $P(s)=-\frac{1}{2\pi i}{\oint }_{\mathrm{\Gamma }}(H(s)-zI{)}^{-1}dz\mathrm{.}$ With $Q(s)=I-P(s)$, define the full gap $g(s)=\min \{\mathrm{\mid }\lambda -\mu \mathrm{\mid }:\lambda \in \sigma (H(s){\mathrm{\mid }}_{\mathrm{ran}P(s)}),\mu \in \sigma (H(s){\mathrm{\mid }}_{\mathrm{ran}Q(s)})\}\mathrm{.}$ If $Q(s)=0$, the full gap is $+\mathrm{\infty }$. Admissible gapped paths have a uniform full gap ${\inf }_{s\in [0,1]}g(s)>0$. Distinct eigenvalues inside the band may cross; its total rank remains $m$.

For a real parameter $T\in \mathbb{R}$, the slow-time propagator $U_T$ is the unique solution of $i\frac{d}{ds}{U}_T(s)=TH(s)U_T(s),U_T(0)=I\mathrm{.}$ Identifying complex coordinates with real and imaginary parts reduces this to a real finite matrix ODE (Linear matrix ODEs have unique global solutions on a fixed interval); differentiating $U_T(s{)}^{\ast }{U}_T(s)$ shows it stays unitary.

An instance also supplies an explicit preparation isometry $A_0:{\mathbb{C}}^m\to V$ with image $\mathrm{ran}P(0)$ and its declared preparation cost. For a rank-one band this means a specified unit vector ${\psi }_0\in \mathrm{ran}P(0)$ and a circuit preparing it; for a rank-$m$ band it is an orthonormal frame, not just the matrix $H(0)$ or the projector $P(0)$. Given an input density operator ${\rho }_{\mathrm{i}\mathrm{n}}$ on ${\mathbb{C}}^m$, the evolved output is ${\rho }_T=U_T(1)A_0{\rho }_{\mathrm{i}\mathrm{n}}{A}_0^{\ast }{U}_T(1{)}^{\ast }$. This is a density operator: for $W=U_T(1)A_0$, $W^{\ast }W=A_0^{\ast }{U}_T(1{)}^{\ast }{U}_T(1)A_0=I$ by isometry and unitarity; hence $W{\rho }_{\mathrm{i}\mathrm{n}}{W}^{\ast }$ is nonnegative by its quadratic form, and $\mathrm{tr}(W{\rho }_{\mathrm{i}\mathrm{n}}{W}^{\ast })=\mathrm{tr}({\rho }_{\mathrm{i}\mathrm{n}}{W}^{\ast }W)=1$ by trace cyclicity (Non-negative and positive operators, The basis-independent trace of an endomorphism of a finite-dimensional vector space, For $A\in M_{m\times n}(F)$ and $B\in M_{n\times m}(F)$, $\mathrm{tr}(AB)=\mathrm{tr}(BA)$). A rank-one input is the pure-state projector (Pure state and ray, Density operator). A supplied target density operator $\sigma$ on $V$ is compared with ${\rho }_T$ by trace distance $D({\rho }_T,\sigma )$ (Trace Distance). A finite output POVM on $V$ and an accepting outcome set may be supplied; its success probability is the sum of the corresponding POVM probabilities (Finite POVMs and their outcome law, POVM outcome weights are probabilities). A finite projective measurement on $V$ may also be supplied, with success probability given by its projective outcome law (Finite projective measurements, Projective measurement outcome weights are probabilities); this is a special case of the POVM interface. The model is mathematical and includes no physical postulate.