Definitions for Renormalization algorithm with graph enhancement

QAlgorithms.Papers.Independent.Intakethird100.P57.ContractionInput

For any natural numbers $N,D$, including zero, and any subset $S\subseteq I_N=\{0,\ldots,N-1\}$, this defines a type of input labels with four distinct forms: a tensor label $(k,b,i,j)$, where $k\in I_N$, $b$ is Boolean, and $i,j\in I_D$; a phase label $(i,j)$, where $i,j\in I_N$; a rotation label $(k,a,b)$, where $k\in I_N$ and $a,b$ are Boolean; and an observable label $(a,b)$, where $a,b:S\to\{\mathrm{false},\mathrm{true}\}$. Phase labels allow equal indices and both orders of each pair. If $N=0$, only observable labels exist, and there is exactly one such label because $S$ is empty; if $D=0$, no tensor labels exist.

QAlgorithms.Papers.Independent.Intakethird100.P57.CyclicInvariant

For every natural number $N$ and every complex vector $v$ indexed by $\{0,1\}^N$, define cyclic invariance to mean that, for every bit string $b$,

$$v\!\left((b_{(k+1)\bmod N})_{k=0}^{N-1}\right)=v(b).$$

No normalization is required. For $N=0$, the shifted string is the unique empty string, so the condition holds for every such vector; for $N=1$, it also holds for every vector because the shift fixes the sole site.

QAlgorithms.Papers.Independent.Intakethird100.P57.DescriptionIndex

For any natural numbers $N,D$, this abbreviates a disjoint union of three kinds of indices: tuples $(k,b,i,j,p)$, where $k<N$, $b,p$ are Booleans, and $i,j<D$; pairs of sites $(j,k)$ satisfying $j<k<N$; and tuples $(k,b,c,p)$, where $k<N$ and $b,c,p$ are Booleans. The three kinds remain distinct even if some of their components coincide. When $N=0$, this index set is empty; when $D=0$, the first kind is empty.

QAlgorithms.Papers.Independent.Intakethird100.P57.EdgeInput

For every triple of natural numbers $N,D,m$ and every family of finite supports $S_r\subseteq\{0,\ldots,N-1\}$ indexed by $r\in\{0,\ldots,m-1\}$, this defines a type of input labels with four distinct kinds: a tensor label $(k,b,i,j)$ with site $0\le k<N$, Boolean value $b$, and indices $0\le i,j<D$; a phase label $(i,j)$ with $0\le i,j<N$; a rotation label $(k,a,b)$ with $0\le k<N$ and Boolean values $a,b$; and a term label $(r,a,b)$ with $0\le r<m$ and Boolean assignments $a,b:S_r\to\{\mathrm{false},\mathrm{true}\}$. Phase labels include ordered pairs and diagonal pairs without restriction. Zero values of $N,D,m$ are allowed and remove whichever labels require an index in the corresponding empty range; an empty support still has one Boolean assignment.

QAlgorithms.Papers.Independent.Intakethird100.P57.EdgeProgram

For every triple of natural numbers $N,D,m$ and family of finite supports $S_r\subseteq\{0,\ldots,N-1\}$ for $0\le r<m$, an edge program consists of a scalar program together with four arbitrary natural-number addresses named the constant coefficient, phase coefficient, norm, and total energy addresses. The scalar program is a finite list with input labels for tensor entries $(k,b,i,j)$, ordered phase pairs $(i,j)$, rotation entries $(k,a,b)$, and term entries $(r,a,b)$, where site indices are below $N$, tensor matrix indices are below $D$, Boolean values are arbitrary, and term assignments are Boolean functions on $S_r$. Its instructions may introduce complex constants, read inputs, add, multiply, divide, conjugate, form $\exp(i\operatorname{Re}z)$, or call a supplied function on a family of $N$ complex $2\times2$ matrices assembled from memory. Execution begins with all addresses zero and writes successive results at successive addresses starting at zero; unwritten addresses read as zero, and division by zero fails. The structure alone gives no equations for its four designated addresses, no bounds placing them within the instruction list, and no guarantee of successful execution. Zero size parameters and empty supports are permitted.

QAlgorithms.Papers.Independent.Intakethird100.P57.EnhancementInput

For every natural number $N$, including zero, and every subset $S\subseteq I_N=\{0,\ldots,N-1\}$, this defines a type of input labels with three distinct forms: a phase label $(i,j)$ for any $i,j\in I_N$; a rotation label $(k,a,b)$ for any $k\in I_N$ and Boolean $a,b$; and an observable label $(a,b)$ for any Boolean assignments $a,b$ on $S$. Phase labels include equal indices and both orders of each pair. If $N=0$, $S$ is empty and the type consists of the single observable label given by the two empty assignments.

QAlgorithms.Papers.Independent.Intakethird100.P57.EnhancementProgram

For every natural number $N$ and finite subset $S\subseteq\{0,\ldots,N-1\}$, an enhancement program consists of a scalar program and a natural-number output address. Its scalar program is a finite list of instructions whose input labels are either an ordered pair of sites $(i,j)$ for a phase, a site and two Boolean values $(k,a,b)$ for a rotation entry, or two Boolean assignments $(a,b)$ on $S$ for an observable entry. Instructions may introduce an arbitrary complex constant, read an input, add, multiply, divide, conjugate, form $\exp(i\operatorname{Re}z)$, or call a supplied function on a family of $N$ complex $2\times2$ matrices whose entries are taken from memory. Execution starts with every memory address equal to zero and writes successive results at addresses $0,1,\ldots$; division by zero makes execution fail. Operand addresses are unrestricted, so an address not yet written reads as zero. The structure imposes no condition that the output address lies within the instruction list or that execution succeeds. Both $N=0$ and $S=\varnothing$ are permitted.

QAlgorithms.Papers.Independent.Intakethird100.P57.ExactProductEvaluator

For every natural number $N$, every complex vector $\psi$ indexed by Boolean strings of length $N$, and every function $\mathrm{base}$ from families of $N$ complex $2\times2$ matrices to $\mathbb C$, the predicate says that, for every such family $F=(F_k)_{k=0}^{N-1}$, $\mathrm{base}(F)=\langle\psi,P_F\psi\rangle$, where matrix action is ordinary matrix–vector multiplication and $(P_F)_{a,b}=\prod_{k=0}^{N-1}(F_k)_{a_k,b_k}$ for Boolean strings $a,b$. There is no normalization assumption on $\psi$ and no restriction on the matrices $F_k$. For $N=0$, the product is empty and equals $1$, so the required value is the squared modulus of the sole coordinate of $\psi$.

QAlgorithms.Papers.Independent.Intakethird100.P57.MarginalProgram.Correct

For any natural numbers $N,D$, subset $S\subseteq I_N=\{0,\ldots,N-1\}$, program record $p$ with scalar code and addresses $\operatorname{stage},\operatorname{cross},\operatorname{norm},\operatorname{entry},\operatorname{expectation}$, arbitrary complex $D\times D$ matrices $A_k(b)$, real weights $w_{ij}$ satisfying symmetry, zero diagonal, and $0\le w_{ij}<2\pi$, unitary complex $2\times2$ matrices $V_k$, and arbitrary complex matrix $O$ indexed by Boolean assignments on $S$, this defines the following predicate. Write $\mathcal B_N=\{\mathrm{false},\mathrm{true}\}^{I_N}$, let $\epsilon(\mathrm{false})=0$ and $\epsilon(\mathrm{true})=1$, and define

$$\phi_c=\operatorname{tr}\!\left(A_0(c_0)\cdots A_{N-1}(c_{N-1})\right),\qquad Z=\sum_{c\in\mathcal B_N}|\phi_c|^2,$$ $$r_b=\sum_{c\in\mathcal B_N} \left(\prod_{k\in I_N}(V_k)_{b_kc_k}\right) \exp\!\left(i\sum_{\substack{i,j\in I_N\\i<j}} w_{ij}\epsilon(c_i)\epsilon(c_j)\right)\phi_c.$$

For assignments $a,b$ on $S$, let $a\sqcup e$ denote the full assignment equal to $a$ on $S$ and to $e$ on its complement, and set

$$R_{ab}= \left(\sum_{c\in\mathcal B_N}|r_c|^2\right)^{-1} \sum_{e:I_N\setminus S\to\{\mathrm{false},\mathrm{true}\}} r_{a\sqcup e}\,\overline{r_{b\sqcup e}}.$$

Define matrices indexed by $I_D\times I_D$ by

$$E_k(u,v)=A_k(u)\otimes\overline{A_k(v)},\qquad T_k(a,b)= \begin{cases} E_k(a_k,b_k),&k\in S,\\[2mm] \displaystyle\sum_{u\in\{\mathrm{false},\mathrm{true}\}} \exp\!\left(i\epsilon(u)\sum_{j\in S} w_{jk}\bigl(\epsilon(a_j)-\epsilon(b_j)\bigr)\right)E_k(u,u), &k\notin S, \end{cases}$$

where the bar on a matrix means entrywise conjugation, and put $P_t(a,b)=T_0(a,b)\cdots T_{t-1}(a,b)$, with $P_0(a,b)$ the identity. The predicate holds exactly when there exists a memory function $m:\mathbb N\to\mathbb C$ obtained by successfully executing all scalar instructions, starting with every address zero and writing the result of instruction number $q$ to address $q$, such that, writing $L$ for the code length,

$$m(\operatorname{norm})=Z,\qquad \operatorname{norm}<L,\qquad \operatorname{expectation}<L,\qquad m(\operatorname{expectation})=\operatorname{tr}(OR),$$

and, for every $a,b$, every $0\le t\le N$, and every $i,j\in I_D\times I_D$,

$$\operatorname{stage}(a,b,t,i,j)<L,\qquad m(\operatorname{stage}(a,b,t,i,j))=(P_t(a,b))_{ij},$$

while, for every $a,b$,

$$\operatorname{cross}(a,b)<L,\qquad m(\operatorname{cross}(a,b))=\operatorname{tr}(P_N(a,b)),\qquad \operatorname{entry}(a,b)<L,\qquad m(\operatorname{entry}(a,b))=R_{ab}.$$

Execution reads tensor entries, complex embeddings of $w_{ij}$, entries of $V_k$, and entries of $O$; constants, addition, multiplication, conjugation, and the operation $z\mapsto e^{i\operatorname{Re}z}$ have their usual meanings; division fails if its denominator is zero; and every external product call returns zero. Operand addresses need not refer to earlier instructions, so an address not yet written still contains zero. No nonzero-vector hypothesis is part of this predicate: the inverse in $R$ is defined to be zero when its denominator is zero. Zero dimensions and empty $S$ are allowed; empty products are identities, and empty sums are zero.

QAlgorithms.Papers.Independent.Intakethird100.P57.MarginalProgram

For natural numbers $N,D$, including zero, and a subset $S\subseteq I_N=\{0,\ldots,N-1\}$, this defines a record containing a scalar program and several natural-number addresses. The scalar program is a finite list of instructions that may supply a complex constant, read a tensor, phase, rotation, or observable input label, add or multiply values at two natural-number addresses, divide values at two addresses, conjugate a value, form $e^{i\operatorname{Re}z}$ from a value $z$, or call an external function on a family of complex $2\times2$ matrices whose entries are specified by addresses indexed by $I_N\times\{\mathrm{false},\mathrm{true}\}^2$. Tensor input labels have indices $(k,b,i,j)\in I_N\times\{\mathrm{false},\mathrm{true}\}\times I_D^2$; phase labels have two indices in $I_N$; rotation labels have one site index and two Boolean indices; observable labels have two Boolean assignments on $S$. The record additionally supplies an address $\operatorname{stage}(a,b,t,i,j)$ for every two Boolean assignments $a,b$ on $S$, every integer $0\le t\le N$, and every $i,j\in I_D\times I_D$; addresses $\operatorname{cross}(a,b)$ and $\operatorname{entry}(a,b)$; and addresses $\operatorname{norm}$ and $\operatorname{expectation}$. The record itself imposes no bounds or semantic requirements on these addresses or on instruction operands.

QAlgorithms.Papers.Independent.Intakethird100.P57.PhaseGraph

For any natural number $N$, this defines a structure consisting of real weights $w_{ij}$ for $0\le i,j<N$, together with the requirements that, for every such pair, $w_{ij}=w_{ji}$, $0\le w_{ij}<2\pi$, and, for every $i$, $w_{ii}=0$. Zero weights are permitted. When $N=0$, there are no weights and all these requirements are vacuous.

QAlgorithms.Papers.Independent.Intakethird100.P57.QubitCliffordGate.matrix

For every natural number $N$, this maps each gate description—Hadamard at a site, phase at a site, or controlled-$X$ on distinct control and target sites—to a complex matrix indexed by Boolean assignments $a,b:I_N\to\{\mathrm{false},\mathrm{true}\}$. Writing $\delta_{uv}=1$ when $u=v$ and $0$ otherwise, a Hadamard description at $q$ has entry

$$H_{a_qb_q}\prod_{k\ne q}\delta_{a_kb_k}, \qquad H_{uv}= \begin{cases} -1/\sqrt2,&u=v=\mathrm{true},\\ 1/\sqrt2,&\text{otherwise}; \end{cases}$$

a phase description at $q$ has entry

$$S_{a_qb_q}\prod_{k\ne q}\delta_{a_kb_k}, \qquad S= \begin{pmatrix}1&0\\0&i\end{pmatrix}$$

in the Boolean ordering false, true; and a controlled-$X$ description with control $j$ and distinct target $k$ has entry $1$ exactly when $a_k=b_k\mathbin{\mathrm{xor}}b_j$ and $a_q=b_q$ for every $q\ne k$, and entry $0$ otherwise. The definition covers $N=0$, although there are then no gate descriptions to which it can be applied.

QAlgorithms.Papers.Independent.Intakethird100.P57.QubitCliffordGate

For every natural number $N$, this defines a type of gate descriptions with three forms: a Hadamard description carrying a site $q\in I_N=\{0,\ldots,N-1\}$; a phase description carrying a site $q\in I_N$; and a controlled-$X$ description carrying control and target sites in $I_N$, together with a proof that they are distinct. Thus the controlled-$X$ form cannot use the same site twice. For $N=0$ there are no gate descriptions, and for $N=1$ there are no controlled-$X$ descriptions.

QAlgorithms.Papers.Independent.Intakethird100.P57.RageParameters

For any natural numbers $N,D$, this defines a parameter record consisting of three fields: a family $A_{k,b}$ of arbitrary complex $D\times D$ matrices indexed by $k<N$ and Boolean $b$; a real weight function $w_{jk}$ on pairs of sites satisfying symmetry $w_{jk}=w_{kj}$, zero diagonal $w_{jj}=0$, and $0\le w_{jk}<2\pi$; and a unitary $2\times2$ complex matrix $V_k$, with Boolean row and column indices, for every site $k<N$. No normalization or nonzero condition is imposed on the matrix family. Zero values of $N$ and $D$ are permitted: when $N=0$, all three fields have empty site index sets, and when $D=0$, the tensor matrices have empty row and column index sets.

QAlgorithms.Papers.Independent.Intakethird100.P57.ScalarInstruction.eval

For every natural number $N$, arbitrary input-label type $I$, input-data function $d:I\to\mathbb C$, arbitrary function $B$ from families of $N$ complex $2\times2$ matrices to $\mathbb C$, and total memory function $m:\mathbb N\to\mathbb C$, this defines instruction evaluation with an optional complex result. A constant returns its specified value; an input read at $i$ returns $d(i)$; addition and multiplication at addresses $a,b$ return $m(a)+m(b)$ and $m(a)m(b)$; division returns no result if $m(b)=0$, and otherwise returns $m(a)/m(b)$; conjugation at $a$ returns $\overline{m(a)}$; phase formation at $a$ returns $e^{i\operatorname{Re}m(a)}$; and a product call with address function $F$ returns $B((M_k)_{k<N})$, where $(M_k)_{ab}=m(F(k,a,b))$. Every form except division by zero returns a value. Addresses are unrestricted natural numbers, and no property of $B$ is assumed; when $N=0$, a product call applies $B$ to the empty matrix family.

QAlgorithms.Papers.Independent.Intakethird100.P57.ScalarInstruction

For every natural number $N$ and arbitrary type of input labels $I$, this defines a type of instructions with nine forms: a complex constant $c$; a read of an input label $i\in I$; addition, multiplication, or division specified by two natural-number memory addresses; conjugation or phase formation specified by one natural-number memory address; and a product call specified by a function assigning a natural-number memory address to each triple $(k,a,b)$, where $k<N$ and $a,b$ are Boolean values. These constructors impose no bounds on memory addresses or requirement that referenced addresses have already been written. The definition allows $N=0$ and allows the input-label type to be empty.

QAlgorithms.Papers.Independent.Intakethird100.P57.ScalarProgram.calls

For every natural number $N$, arbitrary input-label type $I$, and scalar program $p$, this defines the number of entries in its instruction list that are product-call instructions, each carrying an address function on triples consisting of a site $k<N$ and two Boolean values. All other instruction forms contribute zero. This counts occurrences in the list without executing the program, including occurrences after an instruction that would fail; an empty program has count zero, and $N=0$ is allowed.

QAlgorithms.Papers.Independent.Intakethird100.P57.ScalarProgram.execute

For every natural number $N$, arbitrary input-label type $I$, program $p$ containing a finite list of scalar instructions, data function $d:I\to\mathbb C$, and arbitrary function $B$ from families of $N$ complex $2\times2$ matrices to $\mathbb C$, this defines execution returning either no result or a total memory function $\mathbb N\to\mathbb C$. Execution starts with instruction counter $0$ and every memory address equal to zero, and processes the list in order. At each instruction, using the current memory $m$, constants and input reads yield their specified constant and $d(i)$; addition, multiplication, division, and conjugation yield the corresponding operations on the addressed values; phase formation at address $a$ yields $e^{i\operatorname{Re}m(a)}$; and a product call with address function $F$ yields $B((M_k)_{k<N})$, where $(M_k)_{ab}=m(F(k,a,b))$. Division fails precisely when its denominator memory value is zero. A successful instruction stores its result at the current counter address and increments the counter; a failure makes the entire execution return no result. On success, only the final memory function is returned. Memory references need not point to earlier instructions: addresses not yet written read as zero. An empty program succeeds with the all-zero memory, and after a successful program of length $q$, every address at least $q$ remains zero.

QAlgorithms.Papers.Independent.Intakethird100.P57.ScalarProgram

For every natural number $N$ and arbitrary input-label type $I$, this defines a program as a structure containing a single field: a finite list of scalar instructions. Each instruction is a complex constant, an input read, addition, multiplication, division, conjugation, phase formation, or a call carrying memory addresses for the entries of $N$ complex $2\times2$ matrices. The structure imposes no validity conditions, memory-address bounds, designated output address, or requirement that operands refer to earlier instructions. An empty instruction list is permitted, including when $N=0$ or $I$ is empty.

QAlgorithms.Papers.Independent.Intakethird100.P57.ScalarProgram.work

For every natural number $N$, arbitrary input-label type $I$, scalar program $p$, and natural number $B$, whose default value is $0$, this defines work by summing a cost of $4N+1+B$ for every product-call instruction and a cost of $1$ for every other instruction. Equivalently, if the instruction list has length $q$ and contains $c$ product-call instructions, the result is $q+c(4N+B)$. The definition counts the entire list without executing it or checking whether execution succeeds. An empty program has work zero; when $N=0$ and $B=0$, every instruction has cost $1$.

QAlgorithms.Papers.Independent.Intakethird100.P57.SingleEdgeEnergyLaw

For natural numbers $N,D$ supplied with $1\le N$ and $1\le D$, choose complex $D\times D$ matrices $A_r^s$, a graph with real weights $w_{rs}=w_{sr}$, $w_{rr}=0$, and $0\le w_{rs}<2\pi$, unitary complex $2\times2$ matrices $V_r$, distinct sites $j,k<N$, and a Hermitian matrix $H$ indexed by $N$-bit strings, and assume the vector $T(b)=\operatorname{tr}(\prod_{r=0}^{N-1}A_r^{b_r})$, with increasing site order, is nonzero. Put

$$\Phi(b)=\sum_{\substack{r<s\\b_r=b_s=1}}w_{rs},\qquad B(b)=e^{i\Phi(b)-iw_{jk}\mathbf1_{\{b_j=b_k=1\}}}T(b),\qquad W(b,a)=\prod_rV_r(b_r,a_r).$$

Let $u$ equal $B$ on strings for which the two selected bits are not both $1$, and zero elsewhere, and let $v$ equal $B$ on the complementary strings and zero elsewhere. Define

$$c_0=\operatorname{Re}\langle u,W^\dagger HWu\rangle+ \operatorname{Re}\langle v,W^\dagger HWv\rangle,\quad c_1=\langle u,W^\dagger HWv\rangle,\quad Z=\|T\|^2,\quad E_z=W(u+zv),$$

and write $R_H(\xi)=\operatorname{Re}\langle\xi,H\xi\rangle/\|\xi\|^2$ and $m(c)=1$ if $c=0$, otherwise $m(c)=-\overline c/|c|$. The definition is the proposition consisting of the following conjunction:

$$\begin{gathered} \forall z\in\mathbb C,\quad |z|=1\ \Longrightarrow\ \left(\|E_z\|^2=Z\ \text{ and }\ R_H(E_z)=\frac{c_0+2\operatorname{Re}(c_1z)}{Z}\right),\\ |m(c_1)|=1,\\ \forall z\in\mathbb C,\quad |z|=1\ \Longrightarrow\ R_H(E_{m(c_1)})\le R_H(E_z),\\ c_1=0\ \Longrightarrow\ \forall z\in\mathbb C,\quad |z|=1\ \Longrightarrow\ R_H(E_z)=\frac{c_0}{Z}. \end{gathered}$$

Here $Z>0$ follows from the supplied nonzero assumption. This declaration defines a proposition; it does not itself assert that the proposition holds. Although the dimension hypothesis states only $N\ge1$, the required distinct sites cannot be supplied when $N=1$.

QAlgorithms.Papers.Independent.Intakethird100.P57.SiteCoordinates

For every natural number $D$, the abbreviation denotes the set of triples $(b,i,j)$, where $b$ is Boolean and $i,j\in\{0,\ldots,D-1\}$. It has $2D^2$ elements and is empty when $D=0$.

QAlgorithms.Papers.Independent.Intakethird100.P57.Tensors

For any natural numbers $N,D$, including zero, this abbreviates the type of families $A=(A_k(s))$, where $k\in\{0,\ldots,N-1\}$, $s\in\{\mathrm{false},\mathrm{true}\}$, and each $A_k(s)$ is a complex $D\times D$ matrix. No conditions are imposed on these matrices. When $N=0$, the family has no sites; when $D=0$, each matrix has no entries.

QAlgorithms.Papers.Independent.Intakethird100.P57.bitReal

For every Boolean value $b$, define its real-valued representative by

$$\operatorname{bitReal}(b)= \begin{cases} 1,&b=\mathrm{true},\\ 0,&b=\mathrm{false}. \end{cases}$$

QAlgorithms.Papers.Independent.Intakethird100.P57.cliffordMatrix

For every natural number $N$ and finite list of gate descriptions $[g_0,\ldots,g_{\ell-1}]$, this defines the complex matrix on Boolean assignments to $I_N=\{0,\ldots,N-1\}$ to be

$$M_{g_{\ell-1}}\cdots M_{g_1}M_{g_0}.$$

Here a Hadamard gate at $q$ applies $\frac1{\sqrt2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}$ at $q$ and identities elsewhere; a phase gate at $q$ applies $\operatorname{diag}(1,i)$ at $q$ and identities elsewhere; and a controlled-$X$ gate on distinct control $j$ and target $k$ is the permutation matrix sending an input assignment $b$ to the assignment with target bit $b_k\mathbin{\mathrm{xor}}b_j$ and all other bits unchanged. Boolean matrix indices are ordered false, true. An empty list gives the identity matrix, including when $N=0$, when the assignment space has one element and every such list is empty.

QAlgorithms.Papers.Independent.Intakethird100.P57.connectedZ

For every natural number $N$, every complex vector $v$ indexed by bit strings $b\in\{0,1\}^N$, and any sites $i,j\in\{0,\ldots,N-1\}$, define

$$\operatorname{connectedZ}(v,i,j) = \sum_b(-1)^{b_i+b_j}|v(b)|^2 - \left(\sum_b(-1)^{b_i}|v(b)|^2\right) \left(\sum_b(-1)^{b_j}|v(b)|^2\right).$$

There is no normalization assumption on $v$, and $i$ and $j$ may coincide; when $i=j$, this expression is $\sum_b|v(b)|^2-\bigl(\sum_b(-1)^{b_i}|v(b)|^2\bigr)^2$. For $N=0$, there are no choices of $i$ or $j$.

QAlgorithms.Papers.Independent.Intakethird100.P57.contractionData

For natural numbers $N,D$, a subset $S\subseteq I_N=\{0,\ldots,N-1\}$, arbitrary complex $D\times D$ matrices $A_k(b)$ for each site $k\in I_N$ and Boolean $b$, a real weight function $w:I_N\times I_N\to\mathbb R$ satisfying $w_{ij}=w_{ji}$, $w_{ii}=0$, and $0\le w_{ij}<2\pi$, a unitary complex $2\times2$ matrix $V_k$ at each site, and an arbitrary complex matrix $O$ indexed by Boolean assignments on $S$, this defines a complex-valued function on four kinds of input labels: a tensor label $(k,b,i,j)$ returns $(A_k(b))_{ij}$; a phase label $(i,j)$ returns the real number $w_{ij}$, regarded as complex; a rotation label $(k,a,b)$ returns $(V_k)_{ab}$; and an observable label $(a,b)$ returns $O_{ab}$. No normalization of the tensors or Hermiticity of $O$ is required, and $N,D$ may be zero.

QAlgorithms.Papers.Independent.Intakethird100.P57.controlledPhaseAction

For every natural number $N$, any two sites $j,k\in\{0,\ldots,N-1\}$, any complex number $z$, and any complex Euclidean vector $v$ indexed by Boolean assignments on the sites, the definition gives a vector $u$ with amplitudes

$$u_b= \begin{cases} z\,v_b,&b_j=b_k=1,\\ v_b,&\text{otherwise}. \end{cases}$$

There is no requirement that $j\ne k$, that $|z|=1$, or that $v$ be normalized or nonzero. When $j=k$, the condition is simply $b_j=1$; when $N=0$, no sites $j,k$ can be supplied.

QAlgorithms.Papers.Independent.Intakethird100.P57.crossMarginal

For any natural numbers $N,D$, complex $D\times D$ matrices $A_k(t)$ indexed by sites and bits, real weights $w_{j k}$ satisfying symmetry, zero diagonal, and $0\le w_{j k}<2\pi$, and a finite subset $S\subseteq\{0,\ldots,N-1\}$, define a complex matrix $C$ whose rows and columns are bit assignments $a,b:S\to\{0,1\}$ by

$$C_{a,b}=\operatorname{tr}\!\left(T_0^{a,b}\cdots T_{N-1}^{a,b}\right), \qquad T_k^{a,b}= \begin{cases} A_k(a_k)\otimes\overline{A_k(b_k)},&k\in S,\\[2mm] \displaystyle\sum_{t\in\{0,1\}} e^{\,\mathrm{i}t\sum_{j\in S}w_{j k}(a_j-b_j)} \bigl(A_k(t)\otimes\overline{A_k(t)}\bigr),&k\notin S. \end{cases}$$

Bits in the exponential mean $0$ or $1$, conjugation is entrywise, $(B\otimes\overline C)_{(p,q),(r,s)}=B_{p,r}\overline{C_{q,s}}$, and the product is taken in increasing site order. This definition includes $S=\varnothing$, when there is one row and one column. For $N=0$, the product is the identity on the $D^2$ pair indices, so that sole entry is $D^2$; for $D=0$, every entry is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P57.cyclicBlock

For natural numbers $N,\ell$ and a starting index $s\in\{0,\ldots,N-1\}$, this defines the finite set

$$\left\{i\in\{0,\ldots,N-1\}:\exists t\in\mathbb N,\ t<\ell\ \text{and}\ i=(s+t)\bmod N\right\}.$$

Repeated visits contribute only one element. In particular, the set is empty when $\ell=0$, and it contains all $N$ indices when $\ell\ge N$. Supplying a starting index requires $N>0$; there is no starting index when $N=0$.

QAlgorithms.Papers.Independent.Intakethird100.P57.describe

For any natural numbers $N,D$ and any parameter record $p=(A,w,V)$, where $A_{k,b}$ are complex $D\times D$ matrices, $w$ is a symmetric real weight function with zero diagonal and $0\le w_{jk}<2\pi$, and each $V_k$ is a unitary Boolean-indexed $2\times2$ complex matrix, this defines a real-valued function on a disjoint union of tensor-entry indices, strictly ordered site pairs, and rotation-entry indices. At a tensor-entry index $(k,b,i,j,\mathrm{part})$, its value is $\operatorname{Re}(A_{k,b})_{ij}$ if $\mathrm{part}=\mathrm{false}$, and $\operatorname{Im}(A_{k,b})_{ij}$ if $\mathrm{part}=\mathrm{true}$. At a site-pair index $(j,k)$ with $j<k$, its value is $w_{jk}$. At a rotation-entry index $(k,b,c,\mathrm{part})$, its value is $\operatorname{Re}(V_k)_{bc}$ if $\mathrm{part}=\mathrm{false}$, and $\operatorname{Im}(V_k)_{bc}$ if $\mathrm{part}=\mathrm{true}$. All sites lie below $N$, tensor row and column indices lie below $D$, and the remaining indices are Booleans. Zero dimensions are allowed; the function has an empty domain when $N=0$, and has no tensor-entry coordinates when $D=0$.

QAlgorithms.Papers.Independent.Intakethird100.P57.diagonalTensors

For arbitrary natural numbers $N,m$, coefficients $c_t\in\mathbb C$ indexed by $t<m$, and complex numbers $r(t,k,b)$ indexed by $t<m$, $k<N$, and Boolean $b$, this defines a family of $m\times m$ matrices $A_{k,b}$ whose entries are

$$(A_{k,b})_{s,t}= \begin{cases} c_t\,r(t,k,b),&s=t\text{ and }k=0,\\ r(t,k,b),&s=t\text{ and }k\ne0,\\ 0,&s\ne t. \end{cases}$$

No positivity or normalization is required. When $N=0$, there are no site matrices; when $m=0$, every site matrix has an empty row and column index set.

QAlgorithms.Papers.Independent.Intakethird100.P57.edgeBackground

For all natural numbers $N,D$, let $A_i(t)$ be arbitrary complex $D\times D$ matrices indexed by sites and Boolean values, and let $G$ specify real weights $w_{i\ell}$ satisfying $w_{i\ell}=w_{\ell i}$, $w_{ii}=0$, and $0\le w_{i\ell}<2\pi$. Given any sites $j,k$, the definition gives the complex Euclidean vector $u$, indexed by Boolean assignments $b$ on all sites, with amplitudes

$$u_b= \exp\!\left( \mathrm i\left[\sum_{i<\ell}w_{i\ell}b_i b_\ell-w_{jk}b_jb_k\right] \right) \operatorname{tr}\!\left(A_0(b_0)\cdots A_{N-1}(b_{N-1})\right),$$

where Boolean values are identified with $0,1$. No local rotation matrices, normalization, or nonzero hypothesis occur. The sites need not be distinct; when $j=k$, $w_{jj}=0$, so the subtracted term is zero. The definition permits $D=0$, in which case the traces and vector are zero; when $N=0$, no sites $j,k$ exist.

QAlgorithms.Papers.Independent.Intakethird100.P57.edgeCoefficients

For natural numbers $N,D$, choose complex $D\times D$ matrices $A_r^s$ for every site $r<N$ and bit $s$, real graph weights $w_{rs}=w_{sr}$ with $w_{rr}=0$ and $0\le w_{rs}<2\pi$, unitary complex $2\times2$ matrices $V_r$, sites $j,k<N$, and an arbitrary complex matrix $H$ indexed by $N$-bit strings. Set

$$T(b)=\operatorname{tr}\!\left(\prod_{r=0}^{N-1}A_r^{b_r}\right),\qquad \Phi(b)=\sum_{\substack{r<s\\b_r=b_s=1}}w_{rs},\qquad B(b)=e^{i\Phi(b)-iw_{jk}\mathbf1_{\{b_j=b_k=1\}}}T(b),$$

where the matrix product is in increasing site order. Define $W(b,a)=\prod_rV_r(b_r,a_r)$, $H_V=W^\dagger HW$, and vectors $u,v$ by $u(b)=B(b)$ when $b_j,b_k$ are not both $1$, and $u(b)=0$ otherwise, while $v(b)=B(b)$ when both bits are $1$, and $v(b)=0$ otherwise. The definition returns the real–complex pair

$$\left( \operatorname{Re}\langle u,H_Vu\rangle+ \operatorname{Re}\langle v,H_Vv\rangle,\; \langle u,H_Vv\rangle \right).$$

No Hermitian assumption on $H$, distinctness assumption on $j,k$, or normalization or nonzero assumption on $T$ is imposed; $D=0$ is included, whereas $N=0$ supplies no required sites.

QAlgorithms.Papers.Independent.Intakethird100.P57.edgeData

For arbitrary natural numbers $N,D,m$, finite supports $S_r\subseteq\{0,\ldots,N-1\}$ for $0\le r<m$, a family $A_{k,b}$ of complex $D\times D$ matrices for sites $0\le k<N$ and Boolean values $b$, a real weight array $w$ satisfying $w_{ij}=w_{ji}$, $w_{ii}=0$, and $0\le w_{ij}<2\pi$, a family of unitary complex $2\times2$ matrices $V_k$, and arbitrary complex matrices $T_r$ indexed by Boolean assignments on $S_r$, this defines a complex-valued function on the four kinds of edge-input labels: a tensor label $(k,b,i,j)$ returns $(A_{k,b})_{i,j}$; a phase label $(i,j)$ returns $w_{ij}$ viewed as a complex number; a rotation label $(k,a,b)$ returns $(V_k)_{a,b}$; and a term label $(r,a,b)$ returns $(T_r)_{a,b}$. No nonzero or normalization condition is imposed on $A$, and no Hermitian condition is imposed on the terms. All three size parameters may be zero, with labels requiring an unavailable index absent.

QAlgorithms.Papers.Independent.Intakethird100.P57.edgePart

For every natural number $N$, any complex Euclidean vector $v$ indexed by Boolean assignments on $N$ sites, any sites $j,k$, and any Boolean value $\mathrm{on}$, the definition gives the vector $u$ with amplitudes

$$u_b= \begin{cases} v_b,&(b_j\land b_k)=\mathrm{on},\\ 0,&(b_j\land b_k)\ne\mathrm{on}. \end{cases}$$

Thus $\mathrm{on}=1$ retains precisely assignments with both selected bits equal to $1$, while $\mathrm{on}=0$ retains precisely assignments where at least one selected bit is $0$. No normalization or nonzero assumption is imposed, and the retained vector is not rescaled. The sites may coincide, in which case the retained condition is $b_j=\mathrm{on}$; for $N=0$, no sites $j,k$ can be supplied.

QAlgorithms.Papers.Independent.Intakethird100.P57.edgeTrial

For natural numbers $N,D$, let $A_r^s$ be a complex $D\times D$ matrix for each site $r\in\{0,\ldots,N-1\}$ and bit $s\in\{0,1\}$; let $G$ have real weights $w_{rs}$ satisfying $w_{rs}=w_{sr}$, $w_{rr}=0$, and $0\le w_{rs}<2\pi$; and let $V_r$ be a unitary complex $2\times2$ matrix at every site. For sites $j,k$ and any $z\in\mathbb C$, define

$$T(b)=\operatorname{tr}\!\left(A_0^{b_0}\cdots A_{N-1}^{b_{N-1}}\right),\qquad \Phi(b)=\sum_{\substack{r<s\\b_r=b_s=1}}w_{rs},\qquad B(b)=e^{i\Phi(b)} \begin{cases} e^{-iw_{jk}}T(b),&b_j=b_k=1,\\ T(b),&\text{otherwise}. \end{cases}$$

The defined vector, indexed by bit strings $b\in\{0,1\}^N$, has amplitudes

$$\operatorname{edgeTrial}(b)= \sum_{a\in\{0,1\}^N} \left(\prod_{r=0}^{N-1}V_r(b_r,a_r)\right) \begin{cases} zB(a),&a_j=a_k=1,\\ B(a),&\text{otherwise}. \end{cases}$$

There is no requirement that $j\ne k$, that $|z|=1$, or that $T$ be normalized or nonzero. The dimension $D=0$ is allowed; $N=0$ admits no choices of the required sites $j,k$.

QAlgorithms.Papers.Independent.Intakethird100.P57.emptyPhaseGraph

For every natural number $N$, including zero, this defines real weights $w_{ij}=0$ for all $0\le i,j<N$, together with proofs that the weights are symmetric, have zero diagonal, and satisfy $0\le w_{ij}<2\pi$. When $N=0$, the weight function has no arguments and these requirements are vacuous.

QAlgorithms.Papers.Independent.Intakethird100.P57.enhance

For any natural number $N$, real weights $w_{rs}$ on the sites $0,\ldots,N-1$ satisfying $w_{rs}=w_{sr}$, $w_{rr}=0$, and $0\le w_{rs}<2\pi$, any family of unitary complex Boolean-indexed matrices $V_k$, and any vector $v$ in the complex Euclidean space indexed by $N$-bit strings, this defines a vector with coordinates

$$(\operatorname{enhance}(w,V,v))_a =\sum_{b:\{0,\ldots,N-1\}\to\{\mathrm{false},\mathrm{true}\}} \left(\prod_{k=0}^{N-1}(V_k)_{a_k,b_k}\right) \exp\!\left(\mathrm{i}\! \sum_{\substack{0\le r<s<N\\ b_r=\mathrm{true},\ b_s=\mathrm{true}}}w_{rs}\right)v_b.$$

No normalization or nonzero condition is imposed on $v$. For $N=0$, the sole coordinate is unchanged.

QAlgorithms.Papers.Independent.Intakethird100.P57.enhancementData

For any natural number $N$, subset $S\subseteq I_N=\{0,\ldots,N-1\}$, real weight function $w:I_N\times I_N\to\mathbb R$ satisfying $w_{ij}=w_{ji}$, $w_{ii}=0$, and $0\le w_{ij}<2\pi$, unitary complex $2\times2$ matrices $V_k$ for $k\in I_N$, and arbitrary complex matrix $O$ indexed by Boolean assignments on $S$, this defines a complex-valued function on input labels: a phase label $(i,j)$ returns $w_{ij}$, regarded as a complex number; a rotation label $(k,a,b)$ returns $(V_k)_{ab}$; and an observable label $(a,b)$ returns $O_{ab}$. No Hermiticity or positivity of $O$ is assumed. Empty $S$ and $N=0$ are allowed; when $N=0$, the function has only the single observable input label.

QAlgorithms.Papers.Independent.Intakethird100.P57.etaAmplitude

For every complex number $d$ and Boolean $b$, this defines the complex number

$$\eta(d,b)= \begin{cases} e^d,&b=\mathrm{true},\\ 1,&b=\mathrm{false}. \end{cases}$$

There is no restriction on $d$.

QAlgorithms.Papers.Independent.Intakethird100.P57.firstBlock

For any natural numbers $N,L$, define the finite set

$$\operatorname{firstBlock}(N,L)=\{i\in\{0,\ldots,N-1\}:i<L\}.$$

Thus it contains the first $\min(N,L)$ sites; it is empty if $N=0$ or $L=0$, and it contains every site if $L\ge N$.

QAlgorithms.Papers.Independent.Intakethird100.P57.graphAngle

For any type $\iota$ equipped with a finite enumeration and a linear order, any real function $w:\iota\times\iota\to\mathbb R$, and any Boolean assignment $b:\iota\to\{\mathrm{false},\mathrm{true}\}$, this defines the real number

$$\sum_{\substack{i,j\in\iota\\ i<j,\ b_i=\mathrm{true},\ b_j=\mathrm{true}}}w_{ij}.$$

Only strictly ordered pairs contribute; no symmetry, sign, diagonal, or boundedness conditions are imposed on $w$. If there are no contributing pairs, including when $\iota$ is empty, the value is zero.

QAlgorithms.Papers.Independent.Intakethird100.P57.graphOp

For any type $\iota$ equipped with a finite enumeration and a linear order, and any real function $w:\iota\times\iota\to\mathbb R$, this defines the complex diagonal matrix indexed by Boolean assignments $a,b:\iota\to\{\mathrm{false},\mathrm{true}\}$, whose entries are

$$G_{a,b}= \begin{cases} \displaystyle \exp\!\left(\mathrm{i}\! \sum_{\substack{r,s\in\iota\\ r<s,\ b_r=\mathrm{true},\ b_s=\mathrm{true}}}w_{rs}\right),&a=b,\\ 0,&a\ne b. \end{cases}$$

No symmetry, sign, or boundedness condition is required of $w$. If $\iota$ is empty, this is the one-entry matrix with entry $1$.

QAlgorithms.Papers.Independent.Intakethird100.P57.hadamardMatrix

This defines a complex matrix with rows and columns indexed by Booleans whose $(a,b)$-entry is $-1/\sqrt2$ if both $a$ and $b$ are true, and $1/\sqrt2$ otherwise. In the ordering false, true, it is

$$\frac1{\sqrt2} \begin{pmatrix} 1&1\\ 1&-1 \end{pmatrix}.$$

QAlgorithms.Papers.Independent.Intakethird100.P57.joinBits

For every natural number $N$, every finite set $S\subseteq\{0,\ldots,N-1\}$, every Boolean assignment $a$ on $S$, and every Boolean assignment $e$ on its complement, this defines the full $N$-bit string $J_S(a,e)$ by

$$J_S(a,e)_k= \begin{cases} a_k,&k\in S,\\ e_k,&k\notin S. \end{cases}$$

The empty set and the full site set are allowed: in those cases the result uses only $e$ or only $a$, respectively. When $N=0$, the result is the unique empty bit string.

QAlgorithms.Papers.Independent.Intakethird100.P57.localGate

For a natural number $N$, a site $q<N$, and a unitary complex $2\times2$ matrix $U$, this defines the matrix indexed by $N$-bit strings $a,b$ with entries

$$\operatorname{localGate}(b,a) = U(b_q,a_q)\prod_{\substack{r<N\\r\ne q}} \begin{cases} 1,&b_r=a_r,\\ 0,&b_r\ne a_r. \end{cases}$$

Thus it is the product of the matrix $U$ at site $q$ and identity matrices at every other site. The required site $q$ cannot be supplied when $N=0$.

QAlgorithms.Papers.Independent.Intakethird100.P57.localTangent

For every natural number $N$ and arbitrary families $V,\dot V$ of complex $2\times2$ matrices indexed by sites $k<N$, this defines a matrix indexed by bit strings $a,b\in\{0,1\}^N$, with entries

$$\sum_{k=0}^{N-1}\prod_{l=0}^{N-1} \begin{cases} (\dot V_l)_{a_l b_l},&l=k,\\ (V_l)_{a_l b_l},&l\ne k. \end{cases}$$

Neither unitarity nor a derivative relation between the two families is assumed. When $N=0$, the sum is empty, so this is the zero matrix on the singleton set of empty bit strings.

QAlgorithms.Papers.Independent.Intakethird100.P57.localTermHamiltonian

For arbitrary natural numbers $N,m$, a family of finite site sets $S_r\subseteq\{0,\ldots,N-1\}$ indexed by $r\in\{0,\ldots,m-1\}$, and arbitrary complex matrices $T_r$ indexed by pairs of Boolean assignments on $S_r$, this definition gives the complex matrix $H$ indexed by Boolean strings of length $N$ whose entries are $H_{a,b}=\sum_{r=0}^{m-1}\widetilde T_{r;a,b}$, where $\widetilde T_{r;a,b}=(T_r)_{a|_{S_r},b|_{S_r}}$ if $a$ and $b$ agree at every site outside $S_r$, and $\widetilde T_{r;a,b}=0$ otherwise. There are no Hermitian, positivity, normalization, or support-size assumptions. When $m=0$, the sum is the zero matrix; when $S_r=\varnothing$, its matrix has one entry and its extension is that entry times the identity. The definition also permits $N=0$.

QAlgorithms.Papers.Independent.Intakethird100.P57.minimizingPhase

For every complex number $c$, this defines the complex number

$$\operatorname{minimizingPhase}(c)= \begin{cases} 1,&c=0,\\[2pt] -\dfrac{\overline c}{|c|},&c\ne0, \end{cases}$$

where the real modulus in the denominator is regarded as a complex number. The zero case is explicitly assigned the value $1$.

QAlgorithms.Papers.Independent.Intakethird100.P57.mpsTangent

For arbitrary natural numbers $N,D$ and two families $A,\dot A$ assigning a complex $D\times D$ matrix to each site $k<N$ and Boolean value, this defines a vector indexed by bit strings $b\in\{0,1\}^N$, with amplitudes

$$\sum_{k=0}^{N-1}\operatorname{tr}\!\left( \prod_{l=0}^{N-1} \begin{cases} \dot A_l(b_l),&l=k,\\ A_l(b_l),&l\ne k \end{cases}\right),$$

where matrix factors occur in increasing site order. No relation between $A$ and $\dot A$, such as being values and derivatives of a path, is required. For $N=0$ the sum is empty and the vector is zero; for $D=0$ every trace is zero, so the vector is also zero.

QAlgorithms.Papers.Independent.Intakethird100.P57.nextSite

For every natural number $N$ and every site $i\in\{0,\ldots,N-1\}$, define $\operatorname{nextSite}(i)$ to be the site whose index is

$$(i+1)\bmod N.$$

The existence of $i$ ensures $N>0$, so the remainder is a valid site. This sends $N-1$ to $0$; for $N=1$, it fixes the sole site, and for $N=0$, there are no inputs.

QAlgorithms.Papers.Independent.Intakethird100.P57.normalizeRageVector

For any type $\iota$ equipped with a finite enumeration and any vector $v\in\mathbb C^\iota$ with its Euclidean norm, this defines

$$(\|v\|:\mathbb C)^{-1}v.$$

There is no assumption that $v\ne0$. For nonzero $v$, the expression divides $v$ by its norm; for $v=0$, it is zero because the inverse of zero is defined to be zero. In particular, an empty index type yields the zero vector.

QAlgorithms.Papers.Independent.Intakethird100.P57.oppositeWeights

For every natural number $L$, this defines a real-valued function on ordered pairs $i,j\in\{0,\ldots,2L-1\}$ by

$$\operatorname{oppositeWeights}(L)(i,j)= \begin{cases} \pi,&i\ne j\text{ and }(i+L)\bmod(2L)=j,\\ 0,&\text{otherwise}. \end{cases}$$

In particular, diagonal entries receive $0$. The case $L=0$ is included as a function on an empty set of pairs, so there are no entries at which the remainder expression is evaluated.

QAlgorithms.Papers.Independent.Intakethird100.P57.overlapCoefficients

For natural numbers $N,D$, choose complex $D\times D$ matrices $A_r^s$, a graph with real weights $w_{rs}=w_{sr}$, $w_{rr}=0$, and $0\le w_{rs}<2\pi$, sites $j,k<N$, and an arbitrary complex vector $\chi$ indexed by $N$-bit strings. Set

$$T(b)=\operatorname{tr}\!\left(\prod_{r=0}^{N-1}A_r^{b_r}\right),\qquad \Phi(b)=\sum_{\substack{r<s\\b_r=b_s=1}}w_{rs},\qquad B(b)=e^{i\Phi(b)-iw_{jk}\mathbf1_{\{b_j=b_k=1\}}}T(b),$$

using increasing site order in the matrix product. Let $u(b)=B(b)$ when $b_j,b_k$ are not both $1$, and $u(b)=0$ otherwise, and let $v(b)=B(b)$ when both bits are $1$, and $v(b)=0$ otherwise. Writing $a=\langle u,\chi\rangle$ and $b'=\langle v,\chi\rangle$, the definition returns

$$\left(|a|^2+|b'|^2,\;a\,\overline{b'}\right)\in\mathbb R\times\mathbb C.$$

There is no distinctness requirement on $j,k$, no normalization or nonzero requirement on either vector, and no positivity requirement on $D$; $N=0$ admits no choices of the required sites.

QAlgorithms.Papers.Independent.Intakethird100.P57.overlapObjective

For every natural number $N$ and arbitrary complex vectors $v,\chi$ indexed by bit strings in $\{0,1\}^N$, this defines the real number

$$\operatorname{overlapObjective}(v,\chi) =\frac{|\langle v,\chi\rangle|^2}{\|v\|^2}.$$

Neither vector is assumed normalized or nonzero. In particular, the value is $0$ when $v=0$, because division by zero in the real numbers is assigned the value $0$. The case $N=0$ is included, with a single empty bit string indexing the vectors.

QAlgorithms.Papers.Independent.Intakethird100.P57.plusFactors

For every natural number $N$, this defines the complex-valued function on a site $0\le k<N$ and a Boolean value $s$ by

$$r_k(s)=\frac{1}{\sqrt2}.$$

The value is independent of both arguments. When $N=0$, the function has no site arguments.

QAlgorithms.Papers.Independent.Intakethird100.P57.productOp

For any type $\iota$ equipped with a finite enumeration, and any family $V_k$ of complex matrices whose rows and columns are indexed by Boolean values, this defines a complex matrix $P$ indexed by Boolean assignments $a,b:\iota\to\{\mathrm{false},\mathrm{true}\}$, with entries

$$P_{a,b}=\prod_{k\in\iota}(V_k)_{a_k,b_k}.$$

No unitarity or other condition is imposed on $V_k$. If $\iota$ is empty, there is one Boolean assignment and the sole matrix entry is the empty product $1$.

QAlgorithms.Papers.Independent.Intakethird100.P57.productTensors

For every natural number $N$ and every function $r:\{0,\ldots,N-1\}\times\{\mathrm{false},\mathrm{true}\}\to\mathbb C$, this defines a family of $1\times1$ complex matrices $A_{k,b}$, one for each site $k<N$ and Boolean $b$, by

$$(A_{k,b})_{0,0}=r(k,b).$$

There is no normalization condition on $r$. When $N=0$, the family has no sites.

QAlgorithms.Papers.Independent.Intakethird100.P57.productVec

For any natural number $N$ and any complex values $r_k(s)$, indexed by $0\le k<N$ and Boolean $s$, this defines a vector in the complex Euclidean space indexed by $N$-bit strings, with coordinates

$$v_b=\prod_{k=0}^{N-1}r_k(b_k).$$

No normalization or nonzero condition is imposed on the factors. When $N=0$, there is one bit string and its coordinate is the empty product $1$.

QAlgorithms.Papers.Independent.Intakethird100.P57.rage

For any natural numbers $N,D$, any family of complex $D\times D$ matrices $A_k(s)$ indexed by $0\le k<N$ and Boolean $s$, any real weights $w_{rs}$ satisfying symmetry, zero diagonal, and $0\le w_{rs}<2\pi$, and any family of unitary complex Boolean-indexed matrices $V_k$, this defines a vector indexed by $N$-bit strings with coordinates

$$v_a=\sum_{b:\{0,\ldots,N-1\}\to\{\mathrm{false},\mathrm{true}\}} \left(\prod_{k=0}^{N-1}(V_k)_{a_k,b_k}\right) \exp\!\left(\mathrm{i}\! \sum_{\substack{0\le r<s<N\\ b_r=\mathrm{true},\ b_s=\mathrm{true}}}w_{rs}\right) \operatorname{tr}\!\left(A_0(b_0)\cdots A_{N-1}(b_{N-1})\right).$$

The matrix product inside the trace follows increasing site order. No normalization or nonzero condition is imposed. For $N=0$, the sole coordinate is $D$; for $D=0$, the vector is zero.

QAlgorithms.Papers.Independent.Intakethird100.P57.rageMarginal

For every natural number $N$, every complex vector $v$ indexed by $N$-bit strings, and every finite site set $S\subseteq\{0,\ldots,N-1\}$, this defines a matrix indexed by Boolean assignments $a,b$ on $S$, with entries

$$\rho_{a,b}= \bigl(\|v\|^2\bigr)^{-1} \sum_{e:\{0,\ldots,N-1\}\setminus S\to\{\mathrm{false},\mathrm{true}\}} v\!\left(J_S(a,e)\right) \overline{v\!\left(J_S(b,e)\right)},$$

where $J_S(a,e)$ equals $a$ on $S$ and $e$ outside $S$, and the real number $\|v\|^2$ is regarded as complex before taking its inverse. There is no assumption that $v$ is nonzero or normalized; the inverse of zero is defined to be zero, so $v=0$ produces the zero matrix. Empty and full site sets, including $N=0$, are allowed, with a sum over an empty assignment domain containing its single empty assignment.

QAlgorithms.Papers.Independent.Intakethird100.P57.rageSiteQuadratic

For any type $\iota$ equipped with a finite enumeration of all its elements, any complex matrix $M$ with rows and columns indexed by $\iota$, and any function $x:\iota\to\mathbb C$, the definition gives the complex number

$$q_M(x)=\sum_{i\in\iota}\sum_{j\in\iota}\overline{x_i}\,M_{ij}x_j.$$

There is no Hermitian or positivity assumption on $M$, and no normalization requirement on $x$. If $\iota$ is empty, the value is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P57.rawMarginal

For every natural number $N$, every complex vector $v$ indexed by $N$-bit strings, and every finite site set $S\subseteq\{0,\ldots,N-1\}$, this defines a complex matrix indexed by Boolean assignments $a,b$ on $S$, with entries

$$R_{a,b}= \sum_{e:\{0,\ldots,N-1\}\setminus S\to\{\mathrm{false},\mathrm{true}\}} v\!\left(J_S(a,e)\right) \overline{v\!\left(J_S(b,e)\right)},$$

where $J_S(a,e)$ is the full bit string equal to $a$ on $S$ and to $e$ outside $S$. No normalization or nonzero condition is required on $v$. If $S$ is empty, this is a $1\times1$ matrix with entry $\sum_z|v(z)|^2=\|v\|^2$; if $S$ contains all sites, the complement has one empty assignment and the entries are $v(a)\overline{v(b)}$. These conventions also apply when $N=0$.

QAlgorithms.Papers.Independent.Intakethird100.P57.rawTangent

For arbitrary natural numbers $N,D$, complex $D\times D$ matrix families $A,\dot A$ indexed by a site and a Boolean value, real arrays $w,\dot w$ indexed by two sites, and arbitrary complex $2\times2$ matrix families $V,\dot V$, define

$$T_b=\operatorname{tr}\!\left(\prod_{l=0}^{N-1}A_l(b_l)\right),\qquad \dot T_b=\sum_{k=0}^{N-1}\operatorname{tr}\!\left( \prod_{l=0}^{N-1} \begin{cases}\dot A_l(b_l),&l=k,\\A_l(b_l),&l\ne k,\end{cases}\right),$$

with matrix products in increasing site order, and define

$$\Gamma_w(b)=\sum_{\substack{i<j\\b_i=b_j=1}}w_{ij},\qquad P_{ab}=\prod_{l=0}^{N-1}(V_l)_{a_l b_l},\qquad \dot P_{ab}=\sum_{k=0}^{N-1}\prod_{l=0}^{N-1} \begin{cases}(\dot V_l)_{a_l b_l},&l=k,\$V_l)_{a_l b_l},&l\ne k.\end{cases}$$

The definition returns the vector whose amplitude at (a\in{0,1}^N$ is

$$\sum_{b\in\{0,1\}^N} \left[ \dot P_{ab}e^{i\Gamma_w(b)}T_b+ P_{ab}e^{i\Gamma_w(b)} \bigl(\dot T_b+i\Gamma_{\dot w}(b)T_b\bigr) \right].$$

No symmetry or bounds on $w,\dot w$, unitarity of $V,\dot V$, or derivative relationships between any of the paired inputs are required. The resulting vector is zero if $N=0$ or $D=0$.

QAlgorithms.Papers.Independent.Intakethird100.P57.rayleighEnergy

For every natural number $N$, every complex matrix $H$ indexed by Boolean assignments on $N$ sites, and every complex Euclidean vector $v$ with the same index set, the definition assigns the real number

$$E_H(v)=\frac{\operatorname{Re}\langle v,Hv\rangle}{\|v\|^2} = \frac{\operatorname{Re}\displaystyle\sum_{b,c}\overline{v_b}\,H_{bc}v_c} {\displaystyle\sum_b|v_b|^2}.$$

No Hermitian assumption on $H$ or normalization or nonzero assumption on $v$ is imposed. Division is total, so $E_H(0)=0$. The definition also includes $N=0$, when the index set consists of the unique empty Boolean assignment.

QAlgorithms.Papers.Independent.Intakethird100.P57.replaceSite

For all natural numbers $N,D$, let $A_j(b)$ be arbitrary complex $D\times D$ matrices indexed by a site $j\in\{0,\ldots,N-1\}$ and a Boolean value $b$. Given a site $k$ and any complex-valued function $x$ on triples $(b,i,\ell)$ with Boolean $b$ and $0\le i,\ell<D$, the definition gives a new family $A'$ with entries

$$A'_j(b)_{i\ell} = \begin{cases} x(b,i,\ell),&j=k,\\ A_j(b)_{i\ell},&j\ne k. \end{cases}$$

There are no normalization or nonzero requirements. When $D=0$, the matrices and the coordinate domain are empty; when $N=0$, no site $k$ can be supplied.

QAlgorithms.Papers.Independent.Intakethird100.P57.retainedOp

For every natural number $N$, let $G$ specify real weights $w_{ij}$ on the sites $0,\ldots,N-1$, with $w_{ij}=w_{ji}$, $w_{ii}=0$, and $0\le w_{ij}<2\pi$; let $V_i$ be a unitary complex $2\times2$ matrix for each site; and let $S$ be any finite subset of the sites. Identifying Boolean values with $0,1$, the definition gives a complex matrix $R$, indexed by assignments $a,b:S\to\{0,1\}$, whose entries are

$$R_{ab} = \left(\prod_{i\in S}(V_i)_{a_i b_i}\right) \exp\!\left(\mathrm i\sum_{\substack{i,j\in S\\i<j}}w_{ij}b_i b_j\right).$$

Thus the diagonal phase matrix on the retained sites acts first, followed by the product of their individual unitary matrices. The definition includes $N=0$ and $S=\varnothing$; for empty $S$, there is one assignment and $R$ is the $1\times1$ matrix with entry $1$.

QAlgorithms.Papers.Independent.Intakethird100.P57.siteMap

For all natural numbers $N,D$, let $A_i(t)$ be arbitrary complex $D\times D$ matrices for sites $i$ and Boolean values $t$; let $w_{ij}$ be real weights satisfying $w_{ij}=w_{ji}$, $w_{ii}=0$, and $0\le w_{ij}<2\pi$; let each $V_i$ be a unitary complex $2\times2$ matrix; and choose a site $k$. For each coordinate $c=(t,r,s)$, where $t$ is Boolean and $0\le r,s<D$, define matrices $A^{(c)}$ by retaining $A_i(u)$ when $i\ne k$ and setting

$$A^{(c)}_k(u)_{pq} = \begin{cases} 1,&(u,p,q)=c,\\ 0,&(u,p,q)\ne c. \end{cases}$$

The definition gives the matrix $L$, with rows indexed by Boolean assignments $b$ on all $N$ sites and columns indexed by these coordinates $c$, whose entries are

$$L_{bc} = \sum_{a:\{0,\ldots,N-1\}\to\{0,1\}} \left(\prod_i(V_i)_{b_i a_i}\right) e^{\,\mathrm i\sum_{i<j}w_{ij}a_i a_j} \operatorname{tr}\!\left(A^{(c)}_0(a_0)\cdots A^{(c)}_{N-1}(a_{N-1})\right).$$

No nonzero or normalization assumption is required. For $D=0$, this matrix has no columns; for $N=0$, no site $k$ exists.

QAlgorithms.Papers.Independent.Intakethird100.P57.siteVector

For all natural numbers $N,D$, let $C_D$ be the set of triples $(t,i,j)$ with Boolean $t$ and $0\le i,j<D$. Given any complex matrix $L$ whose rows are indexed by Boolean assignments $b$ on $N$ sites and whose columns are indexed by $C_D$, and any function $x:C_D\to\mathbb C$, the definition gives the complex Euclidean vector with amplitudes

$$(\operatorname{siteVector}(L,x))_b=\sum_{c\in C_D}L_{bc}x_c.$$

There are no normalization or nonzero requirements. For $D=0$, the sum is empty and the vector is zero; for $N=0$, there is a single row indexed by the empty assignment.

QAlgorithms.Papers.Independent.Intakethird100.P57.supportedOp

For any natural number $N$, subset $S\subseteq I_N=\{0,\ldots,N-1\}$, and arbitrary complex matrix $O$ indexed by Boolean assignments on $S$, this defines a complex matrix indexed by full Boolean assignments $a,b:I_N\to\{\mathrm{false},\mathrm{true}\}$, with entries

$$\widetilde O_{ab}= \begin{cases} O_{a|_S,b|_S},&a_k=b_k\text{ for every }k\notin S,\\ 0,&\text{otherwise}. \end{cases}$$

No Hermiticity or positivity of $O$ is assumed. Empty $S$ is allowed: then $O$ has one entry and $\widetilde O$ is that scalar times the identity. If $S=I_N$, the agreement condition outside $S$ is vacuous. For $N=0$, the resulting matrix has one row and one column.

QAlgorithms.Papers.Independent.Intakethird100.P57.traceMPS

For any natural numbers $N,D$ and any family of complex $D\times D$ matrices $A_k(s)$, indexed by sites $0\le k<N$ and Boolean values $s$, this defines a vector $v$ in the complex Euclidean space indexed by Boolean strings $b:\{0,\ldots,N-1\}\to\{\mathrm{false},\mathrm{true}\}$, with coordinates

$$v_b=\operatorname{tr}\!\left(A_0(b_0)A_1(b_1)\cdots A_{N-1}(b_{N-1})\right).$$

The factors occur in increasing site order. No normalization is required. For $N=0$, the product is the $D\times D$ identity, so the sole coordinate is $D$, regarded as a complex number; for $D=0$, every coordinate is zero.

QAlgorithms.Papers.Independent.Intakethird100.P57.transferE

For any natural numbers $N,D$, any collection of complex $D\times D$ matrices $A_k(t)$ indexed by sites $k\in\{0,\ldots,N-1\}$ and bits $t$, any site $k$, and any bits $a,b$, define

$$E_k^{a,b}=A_k(a)\otimes\overline{A_k(b)}, \qquad (E_k^{a,b})_{(p,q),(r,s)} = A_k(a)_{p,r}\,\overline{A_k(b)_{q,s}},$$

where $p,q,r,s\in\{0,\ldots,D-1\}$ and the bar denotes entrywise complex conjugation. This is a matrix indexed by pairs of bond indices. If $D=0$, it is the empty matrix; if $N=0$, no site $k$ can be supplied.

QAlgorithms.Papers.Independent.Intakethird100.P57.transferT

For any natural numbers $N,D$, complex $D\times D$ matrices $A_k(t)$ indexed by sites $k\in\{0,\ldots,N-1\}$ and bits $t$, real weights $w_{j k}$ satisfying $w_{j k}=w_{k j}$, $w_{kk}=0$, and $0\le w_{j k}<2\pi$, a finite set of sites $S$, bit assignments $a,b:S\to\{0,1\}$, and a site $k$, define the matrix

$$T_k^{a,b}= \begin{cases} A_k(a_k)\otimes\overline{A_k(b_k)},&k\in S,\\[2mm] \displaystyle\sum_{t\in\{0,1\}} \exp\!\left(\mathrm{i}\,t\sum_{j\in S}w_{j k}(a_j-b_j)\right) \bigl(A_k(t)\otimes\overline{A_k(t)}\bigr),&k\notin S. \end{cases}$$

Here bits in the exponential are interpreted as real numbers $0$ or $1$, conjugation is entrywise, and the tensor product has entries $(B\otimes\overline C)_{(p,q),(r,s)}=B_{p,r}\overline{C_{q,s}}$. If $S$ is empty, every site uses the second branch with exponential factor $1$. If $D=0$, the resulting matrix is empty; if $N=0$, no site $k$ can be supplied.

QAlgorithms.Papers.Independent.Intakethird100.P57.updatedWeights

For a natural number $N$, a graph with real weights $w_{ab}=w_{ba}$, $w_{aa}=0$, and $0\le w_{ab}<2\pi$, sites $j,k<N$, and any real $\theta$, this defines a real-valued function on ordered pairs of sites by

$$w'_{ab}= \begin{cases} w_{jk}+\theta- 2\pi\left\lfloor\dfrac{w_{jk}+\theta}{2\pi}\right\rfloor, &(a=j\text{ and }b=k)\text{ or }(a=k\text{ and }b=j),\\[5pt] w_{ab},&\text{otherwise}. \end{cases}$$

The sites are not required to be distinct: if $j=k$, the selected entry is the diagonal entry $(j,j)$. The output is a weight function, without an accompanying assertion that it satisfies the graph conditions. For $N=0$, the required sites cannot be supplied.

QAlgorithms.Papers.Independent.Intakethird100.P57.wrapAngle

For every real number $\theta$, this defines

$$\operatorname{wrapAngle}(\theta) =\theta-2\pi\left\lfloor\frac{\theta}{2\pi}\right\rfloor,$$

where the floor is an integer regarded as a real number in the product. The definition applies to all real inputs, including negative values and exact integer multiples of $2\pi$.

QAlgorithms.Papers.Independent.Intakethird100.P57.zMean

For every natural number $N$, every complex vector $v$ indexed by $N$-bit strings, and every site $i<N$, this defines the real number

$$Z_i(v)= \sum_{b\in\{\mathrm{false},\mathrm{true}\}^N} \begin{cases} -|v(b)|^2,&b_i=\mathrm{true},\\ |v(b)|^2,&b_i=\mathrm{false}. \end{cases}$$

Here $|v(b)|^2$ is the complex squared modulus. No normalization or nonzero condition is imposed on $v$, and there is no division by its squared norm; the zero vector gives zero. Although $N$ ranges over all natural numbers, no site $i<N$ exists when $N=0$.