Five-local history-state adiabatic path

Input: A full-list uniform circuit on $n\ge 1$ qubits with $L\ge 1$ listed gates $U_1,\dots ,U_L$, each acting on at most two data qubits, and a rational tolerance $0<\epsilon \le 1$. Circuit products act right to left (Full-list uniform quantum circuit family). The computation is applied to the explicitly supplied all-zero input preparation $A_0$; its wire and preparation cost $C_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}$ is recorded by Circuit-family and computation cost model and Adiabatic path access and resource model. The path-access input also supplies uniform term-time providers for the constructed term list, exact coefficient evaluation, supported time codes and certified call errors required by Discretized adiabatic path simulation schedule.

Output: The explicit finite term list and linearly interpolated Hamiltonian path $H(s)$, together with the supplied initial preparation and resource coordinates, plus the evolution access schedule and data-register output produced by steps 5–7 when the required access providers are available. The schedule uses the declared circuit-family or named-oracle provider for each term call and records its corresponding resource charge.

1. Set $K=\lceil 2L\mathrm{/}\epsilon \rceil$ and $N=L+K$. Append $K$ identity gates, writing $U_{\mathrm{\ell }}=I$ for $L<\mathrm{\ell }\le N$. Use data space $({\mathbb{C}}^2{)}^{\otimes n}$ and an ordered $N$-qubit clock register (Finite registers and their ordered computational bases, Ordered tensor registers, their inner product, and wire permutations). Its legal unary clock basis is $\mathrm{\mid }{c}_{\mathrm{\ell }}⟩=\mathrm{\mid }{1}^{\mathrm{\ell }}{0}^{N-\mathrm{\ell }}⟩,0\le \mathrm{\ell }\le N\mathrm{.}$

2. Let $H_{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}={\sum }_{j=1}^{N-1}\mathrm{\mid }01⟩⟨01{\mathrm{\mid }}_{{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}_j,{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}_{j+1}}$ and $H_{\mathrm{i}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}}={\sum }_{i=1}^n\mathrm{\mid }1⟩⟨1{\mathrm{\mid }}_{{\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}}_i}\otimes \mathrm{\mid }0⟩⟨0{\mathrm{\mid }}_{{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}_1},H_{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}0}={\sum }_{j=1}^N\mathrm{\mid }1⟩⟨1{\mathrm{\mid }}_{{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}_j}\mathrm{.}$ The first tensor factor is the data register. The zero-summand convention applies when an index range is empty.

3. For each transition $\mathrm{\ell }-1\to \mathrm{\ell }$, specify two local clock patterns $a_{\mathrm{\ell }},b_{\mathrm{\ell }}$: if $1<\mathrm{\ell }<N$, use $a_{\mathrm{\ell }}=100,b_{\mathrm{\ell }}=110$ on clock bits $(\mathrm{\ell }-1,\mathrm{\ell },\mathrm{\ell }+1)$; for $\mathrm{\ell }=1$, use $a_1=00,b_1=10$ on bits $(1,2)$; for $\mathrm{\ell }=N$, use $a_N=10,b_N=11$ on bits $(N-1,N)$. For a clock pattern $a$, let $\mathrm{\mid }a⟩⟨a\mathrm{\mid }$ act on those listed clock bits and as identity on all other clock bits. Define the five-local propagation term $h_{\mathrm{\ell }}=\frac{1}{2}[I_{\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}}\otimes (\mathrm{\mid }{a}_{\mathrm{\ell }}⟩⟨a_{\mathrm{\ell }}\mathrm{\mid }+\mathrm{\mid }{b}_{\mathrm{\ell }}⟩⟨b_{\mathrm{\ell }}\mathrm{\mid })-U_{\mathrm{\ell }}\otimes \mathrm{\mid }{b}_{\mathrm{\ell }}⟩⟨a_{\mathrm{\ell }}\mathrm{\mid }-U_{\mathrm{\ell }}^{\ast }\otimes \mathrm{\mid }{a}_{\mathrm{\ell }}⟩⟨b_{\mathrm{\ell }}\mathrm{\mid }],H_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}}={\sum }_{\mathrm{\ell }=1}^N{h}_{\mathrm{\ell }}\mathrm{.}$

4. Put $K_0=n+3N$ and ${\mathrm{\Delta }}_0=1\mathrm{/}(216(N+1{)}^3)$, and choose the positive integer $J=2K_0+4K_0^2\mathrm{/}{\mathrm{\Delta }}_0+1.$ Output the linearly interpolated Hamiltonian $H(s)=(1-s)H_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}+s{H}_{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}},H_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}=H_{\mathrm{i}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}}+H_{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}0}+J{H}_{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}},H_{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}=H_{\mathrm{i}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}}+H_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}}+J{H}_{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}\mathrm{.}$ The term list, integer coefficient $J$, initial preparation, term-access queries, coefficient bits, gate costs, and generator time are explicit coordinates of the path access model (Adiabatic path access and resource model); no spectral projector supplies $A_0$. With $M=n+3N-1$ terms, record the rational certificates $M_1=3N$ for the derivative and $B=M^2{J}^2$ for every sampled commutator sum; the coefficient functions are the exact constants $1,J$ and affine functions $s,1-s$.

5. Set ${\mathrm{\Delta }}_0=1\mathrm{/}[216(N+1{)}^3]$, $\gamma =3{\mathrm{\Delta }}_0\mathrm{/}4$, $B_T=6N\mathrm{/}{\gamma }^2+63N^2\mathrm{/}{\gamma }^3$, and $T=\lceil 4B_T\mathrm{/}\epsilon \rceil$. Set the finite-precision simulation tolerance to ${\epsilon }_{\mathrm{s}\mathrm{i}\mathrm{m}}=\epsilon \mathrm{/}4$. These are finite rational computations.

6. Invoke Discretized adiabatic path simulation schedule on the constructed path-access instance with evolution time $T$ and tolerance ${\epsilon }_{\mathrm{s}\mathrm{i}\mathrm{m}}$. Its output schedule applies the supplied $A_0$ before the path evolution, using each term's declared circuit-family or named-oracle provider. If any required term provider, coefficient evaluation or supported call is unavailable, return no schedule.

7. Declare the data register as the output and the clock register as discarded by the output procedure passed to step 6 (the resulting state is the partial trace over the clock, Partial trace over a finite tensor factor). Record $T$, preparation, simulation calls, coefficient precision, circuit-description and generator costs separately.