Linear Combinations of Unitaries and Taylor Simulation

Let $H$ be a finite-dimensional complex inner-product space and let $u,v\in H$ be unit vectors. Write $P_u=\mathrm{\mid }u⟩⟨u\mathrm{\mid }$ and $P_v=\mathrm{\mid }v⟩⟨v\mathrm{\mid }$ using the Dirac convention of Dirac Notation Conversion, and define their trace distance by $D(P_u,P_v):=\frac{1}{2}\mathrm{\parallel }{P}_u-P_v{\mathrm{\parallel }}_1$, where $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_1$ is the trace norm of Trace Norm. Then $D(P_u,P_v)\le \mathrm{\parallel }u-v\mathrm{\parallel }\mathrm{.}$

Let $A$ be a finite register with distinguished unit computational-basis vector $\mathrm{\mid }{0}_A⟩$ (unit by The computational basis is orthonormal), let $S$ be a finite-dimensional complex-inner-product space, and define the isometry $J:S\to A\otimes S$ by $Jx=\mathrm{\mid }{0}_A⟩\otimes x$. The product inner product makes $J$ an isometry; Finite-dimensional linear isometries satisfy the adjoint identity gives $J^{†}J=I_S$, and An isometric quantum encoding gives an orthogonal code projector makes ${\mathrm{\Pi }}_0:=J{J}^{†}$ the orthogonal projection onto its image. On elementary tensors this is $\mathrm{\mid }{0}_A⟩⟨0_A\mathrm{\mid }\otimes I_S$ by the convention of Tensor products of finitely many linear maps. For a unit vector $\varphi \in A\otimes S$, define its zero-branch vector $q:=J^{†}\varphi$, zero-branch weight $p_0:=\mathrm{\parallel }q{\mathrm{\parallel }}^2$, and complementary-branch weight $p_{\perp }:=\mathrm{\parallel }(I-{\mathrm{\Pi }}_0)\varphi {\mathrm{\parallel }}^2$. When $p_0>0$, the conditional system vector is $q\mathrm{/}\sqrt{p_0}$; when $p_0=0$, no conditional vector is assigned. This definition fixes only the clean-ancilla two-outcome branch rule used on this page; it makes no physical realization claim.

Let $H_A,H_E$ be finite-dimensional complex inner-product spaces and let $X=X^{†}$ be an endomorphism of $H_A\otimes H_E$. Then $\mathrm{\parallel }{\mathrm{Tr}}_{E}X{\mathrm{\parallel }}_1\le \mathrm{\parallel }X{\mathrm{\parallel }}_1\mathrm{.}$ The trace norm and partial trace are those of Trace Norm and Partial trace over a finite tensor factor. If either factor is zero-dimensional, both sides are zero.

With the notation of Clean-ancilla zero-branch postselection, every unit vector $\varphi$ has $p_0,p_{\perp }\in [0,1]$ and $p_0+p_{\perp }=1$. If $p_0>0$, the assigned conditional vector $q\mathrm{/}\sqrt{p_0}$ is a unit vector; if $p_0=0$, then $q=0$ and no conditional vector exists.

Let $S$ be a finite-dimensional complex-inner-product system space and let $M={\sum }_{j=0}^{m-1}{c}_j{V}_j,c_j\in \mathbb{C},$ where $m\ge 1$, each $V_j:S\to S$ is unitary, and $\alpha :={\sum }_{j=0}^{m-1}\mathrm{\mid }{c}_j\mathrm{\mid }>0.$ The finite ordered list $(c_j,V_j{)}_{j<m}$ is an LCU representation of $M$; it is part of the input, and a matrix description alone does not supply circuits for any $V_j$.

Set $q=\lceil {\log }_{2}m\rceil$ and let $A_{\mathrm{i}\mathrm{d}\mathrm{x}}$ be the $q$-qubit index register, with the one-dimensional $q=0$ case allowed. Its computational indices are $0,\dots ,2^q-1$. For $j<m$ put ${\omega }_j=c_j\mathrm{/}\mathrm{\mid }{c}_j\mathrm{\mid }$ when $c_j\mathrm{\ne }0$ and ${\omega }_j=1$ when $c_j=0$. Define the complete SELECT unitary by $S_{\mathrm{s}\mathrm{e}\mathrm{l}}(\mathrm{\mid }j⟩\otimes x)=\{\begin{array}{ll}\mathrm{\mid }j⟩\otimes {\omega }_j{V}_{j}x,& j<m,\\ \mathrm{\mid }j⟩\otimes x,& m\le j<2^q\mathrm{.}\end{array}$ This basis rule fixes every valid and padded index; it extends linearly to all superpositions and acts as identity on any separately declared work register.

A PREPARE access is a fully specified unitary $P$ on the index register and all its declared work registers, with $P\mathrm{\mid }{0}_{\mathrm{a}\mathrm{n}\mathrm{c}}⟩=e^{i\theta }({\sum }_{j<m}\sqrt{\mathrm{\mid }{c}_j\mathrm{\mid }\mathrm{/}\alpha }\mathrm{\mid }j⟩)\otimes \mathrm{\mid }{0}_{\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k}}⟩$ for some phase $\theta \in \mathbb{R}$. The same full unitary, including its action on every nonzero input, determines $P^{†}$; neither an inverse nor a controlled call is implied unless explicitly supplied. The allowed global phase cancels between $P$ and $P^{†}$ in the LCU block. PREPARE work wires are included among the ancillas projected to zero.

For an amplitude-table implementation with $D=2^q$ entries, the dense fixed-width input length, table reading, arithmetic, and coefficient precision are separate costs. If the continuous-rotation QC24 procedure QC24 State Preparation is used, its source theorem QC24 State Preparation Rotation Level Upper Bound gives $O((q+1)D)$ bounded-arity gates and serial depth, with at most $\max \{q-1,0\}$ reusable clean work bits; the table and exact arithmetic costs remain separate. This bound is for continuous rotations. Finite-alphabet synthesis and finite-precision bit-time require separate guarantees. A supplied PREPARE circuit is instead charged by its declared full description, gates, wires, and generation cost.

The LCU block circuit is the complete unitary $W:=P^{†}{S}_{\mathrm{s}\mathrm{e}\mathrm{l}}P$ on all coefficient/work ancillas and the system. The access record separately lists PREPARE, PREPARE${}^{†}$, SELECT, SELECT${}^{†}$, and every controlled form used, together with their query, gate, wire, coefficient-bit, and classical-description costs under Hamiltonian simulation error and resources and Circuit-family and computation cost model. Unspecified access, inverse access, controlled access, and table loading have no free cost. The zero LCU representation is outside this access model because $\alpha =0$ does not define a normalized coefficient state.

Input: An LCU representation and complete PREPARE and SELECT access from LCU decomposition and access model, including their stated work-register and padded-index behavior. The access record must explicitly supply PREPARE${}^{†}$ and its cost; an input without that inverse access is inadmissible.

Procedure: On the coefficient/work ancillas and system, apply PREPARE, then SELECT, then the supplied PREPARE${}^{†}$. Return the complete unitary $W=P^{†}{S}_{\mathrm{s}\mathrm{e}\mathrm{l}}P$. Charge the three calls at their declared costs. No measurement or deterministic state-output guarantee is part of this procedure.

Let $A$ be the full coefficient and PREPARE-work ancilla register in LCU decomposition and access model, let $S$ be the system, and let $J:S\to A\otimes S$ be the clean injection of Clean-ancilla zero-branch postselection. Set ${\mathrm{\Pi }}_0:=J{J}^{†}$; An isometric quantum encoding gives an orthogonal code projector makes this the orthogonal projection onto the image of $J$. Its basis formula is ${\mathrm{\Pi }}_0=\mathrm{\mid }{0}_A⟩⟨0_A\mathrm{\mid }\otimes I_S$, where $\mathrm{\mid }{0}_A⟩$ is unit by The computational basis is orthonormal and the tensor action is that of Tensor products of finitely many linear maps. Define $R_0:=I-2{\mathrm{\Pi }}_0\mathrm{.}$ Thus $R_0$ acts as $-I$ on the clean-ancilla subspace and as $+I$ on its orthogonal complement; it acts identically on the unknown system state. In the convention of Orthogonal-Projector Reflection and Access Contract, $R_0=S_{{\mathrm{\Pi }}_0}$, while that item writes $R_{{\mathrm{\Pi }}_0}=2{\mathrm{\Pi }}_0-I=-R_0$. These signs are distinct inside controlled blocks: if only the controlled $R_{{\mathrm{\Pi }}_0}$ is supplied, implement controlled $R_0$ with the exact control-$Z$ correction of Projector Reflections Are Hermitian Unitary Involutions and charge that gate. A projector-reflection call and its controlled form are explicit access operations, not free consequences of a state vector.

Let $H={\sum }_{j=0}^{m-1}{a}_j{V}_j$ on a finite-dimensional complex-inner-product space, with $m\ge 1$, real $a_j$ and Hermitian unitaries $V_j$, and put $\mathrm{\Lambda }={\sum }_j\mathrm{\mid }{a}_j\mathrm{\mid }$. For real $h$ and integer $K\ge 0$, define $M_K(h):={\sum }_{k=0}^K\frac{(-ihH{)}^k}{k!}\mathrm{.}$ Its ordered product-term LCU list has one empty tuple for $k=0$, with coefficient $1$ and unitary $I$, and for each $1\le k\le K$ and ordered tuple $(j_1,\dots ,j_k)$ has coefficient $c_{k,\mathbf{j}}:=\frac{(-i{)}^k{h}^k{a}_{j_1}\cdots a_{j_k}}{k!}$ and unitary $V_{j_1}\cdots V_{j_k}$. Distinct tuples remain distinct entries even if their product unitaries coincide. The absolute coefficient sum of this listed representation is $s_K(h):={\sum }_{k=0}^K\frac{(\mathrm{\Lambda }\mathrm{\mid }h\mathrm{\mid }{)}^k}{k!}\mathrm{.}$

Whenever $s_K(h)\le 2$, append two distinct identity entries with coefficients $+q$ and $-q$, where $q=(2-s_K(h))\mathrm{/}2$. They leave the represented matrix unchanged and raise the listed absolute coefficient sum to exactly $2$. If $q=0$, the zero-weight entries may be retained with zero preparation amplitude or omitted. The polynomial, coefficient phases, ordered tuple index, and any numerical precision used to generate them are part of the finite input/access record; no term product, coefficient table, PREPARE circuit, or SELECT circuit is free.

Let $M={\sum }_{j<m}{c}_j{V}_j$ be a finite LCU representation with $\alpha ={\sum }_j\mathrm{\mid }{c}_j\mathrm{\mid }>0$, let $W$ be the unitary from LCU block encoding, and let $Jx=\mathrm{\mid }{0}_{\mathrm{a}\mathrm{n}\mathrm{c}}⟩\otimes x$ include every coefficient and PREPARE-work ancilla. Then $J^{†}WJ=M\mathrm{/}\alpha \mathrm{.}$ For every unit input vector $\psi$, the clean-ancilla zero-branch vector is $M\psi \mathrm{/}\alpha$ and its postselection weight, under Clean-ancilla zero-branch postselection, is $\mathrm{\parallel }M\psi {\mathrm{\parallel }}^2\mathrm{/}{\alpha }^2$. If this weight is positive, the conditional system vector is $M\psi \mathrm{/}\mathrm{\parallel }M\psi \mathrm{\parallel }$; if it is zero, the conditional vector is undefined.

Input: A unitary $W$ on coefficient/work ancillas and system, explicit inverse access to $W$, the reflection $R_0=I-2{\mathrm{\Pi }}_0$ of Oblivious ancilla reflection, and an operator $M$ satisfying $J^{†}WJ=M\mathrm{/}2$ for the clean injection $Jx=\mathrm{\mid }{0}_{\mathrm{a}\mathrm{n}\mathrm{c}}⟩\otimes x$.

Procedure: Chronologically apply $W$, then $R_0$, then $W^{†}$, then $R_0$, then $W$. The resulting unitary is $C:=-W{R}_0{W}^{†}{R}_{0}W\mathrm{.}$ The scalar $-1$ is an explicit phase correction; it is not omitted when this procedure is controlled. The circuit uses two $W$ calls, one $W^{†}$ call, and two reflection calls. A controlled invocation requires the declared controlled forward, inverse, reflection, and phase-correction gates. This procedure makes no measurement and asserts no exact ancilla cleanup for a nonunitary or approximate $M$; the robust clean-input guarantee is proved separately.

Let $H={\sum }_{j<m}{a}_j{V}_j$ be a finite-dimensional self-adjoint Hamiltonian on a nonzero complex-inner-product space, with $m\ge 1$, written with real $a_j$ and Hermitian unitaries $V_j$, and set $\mathrm{\Lambda }={\sum }_j\mathrm{\mid }{a}_j\mathrm{\mid }$. For real $h$, integer $K\ge 0$, and $x=\mathrm{\Lambda }\mathrm{\mid }h\mathrm{\mid }$, $\parallel M_K(h)-e^{-ihH}\parallel \le e^x\frac{x^{K+1}}{(K+1)!}\mathrm{.}$ The Taylor coefficient sum $s_K={\sum }_{k=0}^K{x}^k\mathrm{/}k!$ satisfies $s_K\le e^x\le 2$ when $x\le \ln 2$. If $0<\eta <1$ and $x\le \ln 2$, there is a $K=O\text{\hspace{0.17em}⁣}(1+\frac{\log (1\mathrm{/}\eta )}{\log \log (e+1\mathrm{/}\eta )})$ for which the right-hand side is at most $\eta$.

Let $J:S\to A\otimes S$ be the clean-ancilla isometry $Jx=\mathrm{\mid }{0}_{\mathrm{a}\mathrm{n}\mathrm{c}}⟩\otimes x$, let $\mathrm{\Pi }=J{J}^{†}$ be the orthogonal projection onto its image, and set $R_0=I-2\mathrm{\Pi }$. Let $W$ be unitary and assume $J^{†}WJ=M\mathrm{/}2$. For the chronological circuit $W{R}_0{W}^{†}{R}_{0}W$, $J^{†}W{R}_0{W}^{†}{R}_{0}WJ=-\frac{3}{2}M+\frac{1}{2}M{M}^{†}M\mathrm{.}$ If $M$ is unitary, this projected block is $-M$. The identity alone gives no bound on leakage outside the clean-ancilla subspace for a nonunitary $M$.

Let $S$ be a nonzero finite-dimensional system space, let $A$ be the complete clean-ancilla register, let $Jx=\mathrm{\mid }{0}_A⟩\otimes x$, and put $\mathrm{\Pi }=J{J}^{†}$, $R_0=I-2\mathrm{\Pi }$. Suppose $W$ is unitary, $J^{†}WJ=M\mathrm{/}2$, $V:S\to S$ is unitary, and $\mathrm{\parallel }M-V\mathrm{\parallel }\le \eta \le 1\mathrm{/}100$. For $C:=-W{R}_0{W}^{†}{R}_{0}W$ one has the clean-input isometry bound $\mathrm{\parallel }CJ-JV\mathrm{\parallel }\le 10\sqrt{\eta }\mathrm{.}$ The estimate includes residual ancilla leakage; it does not assert exact cleanup for nonzero $\eta$.