Definitions for Characterization of non-local gates

QAlgorithms.Papers.Independent.Intakethird100.P99.CanCbit

For a complex two-qubit matrix $U$, this predicate says that $U$ is unitary and there exist single-qubit unitaries $E,D$, a vector $v\in\mathbb C^{\{\mathrm{false},\mathrm{true}\}}$, and vectors $w_i$ in the same space for each Boolean $i$, such that $\|v\|=1$, $\|w_i\|=1$ for both $i$, and

$$U\bigl((E|i\rangle)\otimes v\bigr)=w_i\otimes(D|i\rangle) \qquad\text{for every }i\in\{\mathrm{false},\mathrm{true}\}.$$

Here $|i\rangle$ is the coordinate vector with value one at $i$ and zero at the other Boolean, matrix action is ordinary matrix-vector multiplication, and $(x\otimes y)_{(a,b)}=x_a y_b$. All norms are Euclidean. The same $E,D,v$ must work for both values of $i$; $w_i$ may depend on $i$.

QAlgorithms.Papers.Independent.Intakethird100.P99.CanHybrid

For a complex two-qubit matrix $U$, this predicate says that $U$ is unitary and there exist fixed single-qubit unitaries $E,F$, a single-qubit unitary $R_j$ for each Boolean $j$, and single-qubit vectors $a_j$ such that $\|a_j\|=1$ for both $j$, $\langle a_{\mathrm{false}},a_{\mathrm{true}}\rangle=0$, and

$$\bigl((I_2\otimes R_j)U\bigr) \bigl((Eq)\otimes(F|j\rangle)\bigr)=a_j\otimes q \qquad\text{for every Boolean }j\text{ and every }q\text{ with }\|q\|=1.$$

The vectors lie in $\mathbb C^{\{\mathrm{false},\mathrm{true}\}}$, norms are Euclidean, the inner product is the standard complex inner product conjugate-linear in its first argument, and $|j\rangle$ is the coordinate basis vector. Matrix actions are ordinary multiplication and tensor-vector coordinates are products. Both $R_j$ and $a_j$ may depend on $j$, but none of the chosen objects may depend on $q$.

QAlgorithms.Papers.Independent.Intakethird100.P99.CanQubit

For a complex two-qubit matrix $U$, this predicate says that $U$ is unitary and there exist fixed single-qubit unitaries $E,R$ and fixed single-qubit vectors $v,w$ of Euclidean norm one such that, for every single-qubit vector $q$ with $\|q\|=1$,

$$\bigl((I_2\otimes R)U\bigr)\bigl((Eq)\otimes v\bigr)=w\otimes q.$$

All single-qubit vectors belong to $\mathbb C^{\{\mathrm{false},\mathrm{true}\}}$, $I_2$ is its identity matrix, and tensor-vector coordinates are products of coordinates. The equality is exact, and the same choices $E,R,v,w$ must work for every unit vector $q$.

QAlgorithms.Papers.Independent.Intakethird100.P99.CanTwoCbits

For a complex two-qubit matrix $U$, this predicate says that $U$ is unitary and there exist single-qubit unitaries $E,F$ and single-qubit vectors $a_{ij},b_{ij}$ for every pair of Booleans $i,j$, such that

$$\|a_{ij}\|=\|b_{ij}\|=1,\qquad \langle a_{i,\mathrm{false}},a_{i,\mathrm{true}}\rangle=0 \quad\text{for every }i,\qquad \langle b_{\mathrm{false},j},b_{\mathrm{true},j}\rangle=0 \quad\text{for every }j,$$

and

$$U\bigl((E|i\rangle)\otimes(F|j\rangle)\bigr)=a_{ij}\otimes b_{ij} \qquad\text{for every }i,j.$$

The vectors lie in $\mathbb C^{\{\mathrm{false},\mathrm{true}\}}$, with the Euclidean norm and the standard complex inner product, conjugate-linear in its first argument. The ket $|i\rangle$ is the corresponding coordinate vector, matrix action is ordinary multiplication, and tensor-vector coordinates are products of coordinates. The unitaries $E,F$ are fixed across all four inputs.

QAlgorithms.Papers.Independent.Intakethird100.P99.CanTwoQubits

For a complex two-qubit matrix $U$, this predicate says that $U$ is unitary and there exist fixed single-qubit unitaries $E,F,R,S$ such that

$$\bigl((R\otimes S)U\bigr)\bigl((Eq)\otimes(Fr)\bigr)=r\otimes q \qquad\text{for all single-qubit vectors }q,r\text{ with }\|q\|=\|r\|=1.$$

The single-qubit space is $\mathbb C^{\{\mathrm{false},\mathrm{true}\}}$, the norms are Euclidean, matrix action is ordinary multiplication, and tensor-vector coordinates are products of coordinates. The equality is exact and the four unitaries must work simultaneously for every such pair $q,r$.

QAlgorithms.Papers.Independent.Intakethird100.P99.ChamberClass

For every real triple $b=(b_0,b_1,b_2)$ and every index $k\in\{0,1,2,3\}$, this predicate is defined by

$$\begin{cases} b_0<\pi/4, & k=0,\\ b_0=\pi/4\ \text{and}\ b_1<\pi/4, & k=1,\\ b_0=\pi/4,\ b_1=\pi/4,\ \text{and}\ |b_2|<\pi/4, & k=2,\\ b_0=\pi/4,\ b_1=\pi/4,\ \text{and}\ |b_2|=\pi/4, & k=3. \end{cases}$$

No additional inequalities on the coordinates are part of this predicate; in particular, it does not separately require $b_1\le b_0$ or $|b_2|\le b_1$.

QAlgorithms.Papers.Independent.Intakethird100.P99.ControlPhase

For every complex matrix $W$ whose rows and columns are indexed by $\{\mathrm{false},\mathrm{true}\}$, and every real number $b$, this predicate means that $0\le b\le\pi/2$ and there exists a complex number $z$ with $|z|=1$ such that the characteristic polynomial of the scalar multiple $zW$ is

$$\chi_{zW}(x)=(x-e^{2ib})(x-e^{-2ib}).$$

The endpoints $b=0$ and $b=\pi/2$ are included. The predicate imposes no separate unitarity or Hermiticity condition on $W$.

QAlgorithms.Papers.Independent.Intakethird100.P99.GateChamber

For a real triple $b=(b_0,b_1,b_2)$, this predicate means precisely that $b_1\le b_0$, $|b_2|\le b_1$, and $b_0\le\pi/4$. All boundary equalities are allowed.

QAlgorithms.Papers.Independent.Intakethird100.P99.HasCanonicalHamiltonian

For a complex $4\times4$ matrix $H$, indexed by pairs of Boolean values, and a real triple $a=(a_0,a_1,a_2)$, this predicate holds precisely when $H$ is Hermitian, $a_1\le a_0$ and $|a_2|\le a_1$, and there exist complex $2\times2$ unitary matrices $A,B$ and complex $2\times2$ Hermitian matrices $K,L$ satisfying

$$(A\otimes B)H(A\otimes B)^\dagger =\sum_{j=0}^{2}a_j\,\sigma_j\otimes\sigma_j+K\otimes I_2+I_2\otimes L,$$

where $(\sigma_0,\sigma_1,\sigma_2)=(X,Y,Z)$ are the usual Pauli matrices, $I_2$ is the identity matrix, and $\dagger$ denotes conjugate transpose. No upper bound on $a_0$, determinant-one restriction on $A,B$, or trace restriction on $K,L$ is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P99.HasContent

For a complex $4\times4$ matrix $U$, indexed by pairs of Boolean values, and a real triple $b=(b_0,b_1,b_2)$, this predicate holds precisely when $U$ is unitary,

$$b_1\le b_0,\qquad |b_2|\le b_1,\qquad b_0\le\frac{\pi}{4},$$

and there exist four complex $2\times2$ unitary matrices $A,B,C,D$ such that

$$U=(A\otimes B)\exp\!\left(-\mathrm{i}\sum_{j=0}^{2}b_j\,\sigma_j\otimes\sigma_j\right)(C\otimes D),$$

where $(\sigma_0,\sigma_1,\sigma_2)=(X,Y,Z)$ are the usual Pauli matrices and $\otimes$ denotes the Kronecker product. The equality is exact, and the four local unitary matrices have no determinant-one requirement.

QAlgorithms.Papers.Independent.Intakethird100.P99.IdealSimulates

For complex $4\times4$ matrices $H,U$, indexed by pairs of Boolean values, and a real number $t$, this predicate holds precisely when $H$ is Hermitian, $U$ is unitary, $t\ge0$, and for every real $\varepsilon>0$ there exist a finite list $(t_j,A_j,B_j)_{j=1}^{m}$, initial matrices $A_0,B_0$, and a complex scalar $z$, such that all $A_j,B_j$, including the initial pair, are complex $2\times2$ unitaries, every $t_j\ge0$, $\sum_{j=1}^{m}t_j=t$, $|z|=1$, and

$$\|V_m-zU\|_{\mathrm{op}}<\varepsilon,\qquad V_0=A_0\otimes B_0,\qquad V_j=(A_j\otimes B_j)\exp(-\mathrm{i}t_jH)V_{j-1}.$$

Here $\|\cdot\|_{\mathrm{op}}$ is the operator norm induced by the Euclidean norm. The list and phase may depend on $\varepsilon$; empty lists, zero durations, and total time $t=0$ are included.

QAlgorithms.Papers.Independent.Intakethird100.P99.IsGate

For any $4\times4$ complex matrix $U$ indexed by pairs of Boolean values, this predicate means that $U$ is unitary: $U^\dagger U=UU^\dagger=I_4$.

QAlgorithms.Papers.Independent.Intakethird100.P99.MoreNonlocal

For arbitrary complex two-qubit matrices $U,V$, this predicate says that for every Hermitian complex two-qubit matrix $H$, $C_H(V)\le C_H(U)$. Here $C_H(W)\in[0,\infty]$ is the infimum of all nonnegative real times $t$ such that $W$ is unitary and, for every $\varepsilon>0$, there exist a finite list of nonnegative durations $t_1,\ldots,t_m$ summing exactly to $t$, single-qubit unitaries $A_0,B_0,\ldots,A_m,B_m$, and $z\in\mathbb C$ with $|z|=1$, for which

$$\left\|(A_m\otimes B_m)e^{-it_mH}\cdots (A_1\otimes B_1)e^{-it_1H}(A_0\otimes B_0)-zW\right\|_{\mathrm{op}}<\varepsilon.$$

The norm is the Euclidean operator norm. The list may be empty, giving duration zero and matrix $A_0\otimes B_0$; the protocol and phase may depend on $\varepsilon$. The infimum of an empty set is $+\infty$. Neither $U$ nor $V$ is assumed unitary in the predicate itself, and the comparison includes infinite costs, with $+\infty\le+\infty$ allowed.

QAlgorithms.Papers.Independent.Intakethird100.P99.OneGate

This is the type of unitary $2\times2$ complex matrices indexed by Boolean values: matrices $A$ satisfying $A^\dagger A=AA^\dagger=I_2$. Their determinants are not required to equal $1$.

QAlgorithms.Papers.Independent.Intakethird100.P99.OneMatrix

This is the type of all complex matrices whose rows and columns are indexed by $\{\mathrm{false},\mathrm{true}\}$, hence all $2\times2$ complex matrices, with no further restrictions.

QAlgorithms.Papers.Independent.Intakethird100.P99.OneState

This is the complex Euclidean space of vectors indexed by Boolean values, namely $\mathbb C^2$ with its Euclidean inner product. Vectors are not required to have norm $1$; the zero vector is included.

QAlgorithms.Papers.Independent.Intakethird100.P99.OptimalTask

For arbitrary complex two-qubit matrices $H,U$, a predicate $P$ on such matrices, and a real number $t$, this predicate asserts all of the following simultaneously: $P(U)$ holds; $H$ is Hermitian, $U$ is unitary, and $t\ge0$; for every $\varepsilon>0$ there exist a finite list of nonnegative durations $t_1,\ldots,t_m$ summing exactly to $t$, single-qubit unitaries $A_0,B_0,\ldots,A_m,B_m$, and $z\in\mathbb C$ with $|z|=1$, such that

$$\left\|(A_m\otimes B_m)e^{-it_mH}\cdots (A_1\otimes B_1)e^{-it_1H}(A_0\otimes B_0)-zU\right\|_{\mathrm{op}}<\varepsilon;$$

moreover,

$$C_H(U)=\operatorname{ofReal}(t),\qquad \inf_{V:\,P(V)}C_H(V)=\operatorname{ofReal}(t),\qquad \forall V,\ P(V)\Longrightarrow C_H(U)\le C_H(V).$$

Here $C_H(W)$ is the infimum in $[0,\infty]$ of all nonnegative times admitting the same approximation condition with target $W$, including the requirements that $H$ be Hermitian and $W$ unitary; an empty admissible set has cost $+\infty$. The operator norm is induced by the Euclidean norm. Empty pulse lists are allowed and give duration zero and matrix $A_0\otimes B_0$, while protocols and phases may vary with $\varepsilon$. Finally, $\operatorname{ofReal}(t)$ is $\max(t,0)$ regarded as an extended nonnegative real, which here equals $t$ because nonnegativity is required.

QAlgorithms.Papers.Independent.Intakethird100.P99.PhaseCompatibleContent

For every complex matrix $U$ on two qubits and every real triple $b=(b_0,b_1,b_2)$, this predicate means that

$$b_1\le b_0,\qquad |b_2|\le b_1,\qquad b_0\le\frac{\pi}{4},$$

and there exist four complex $2\times2$ unitary matrices $A,B,C,D$, each with determinant $1$, such that

$$U=(A\otimes B)\exp\!\left(-i\sum_{j=0}^{2}b_j(P_j\otimes P_j)\right)(C\otimes D),$$

where $P_0=X,P_1=Y,P_2=Z$ are the Pauli matrices

$$X=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad Y=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\qquad Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$$

in Boolean basis order $\mathrm{false},\mathrm{true}$. The displayed equality is exact and contains no additional overall phase factor.

QAlgorithms.Papers.Independent.Intakethird100.P99.PulseProtocol

A pulse protocol consists of two initial complex $2\times2$ unitary matrices $A_0,B_0$ and a finite ordered list of triples $(t_j,A_j,B_j)$, where every $t_j$ is a nonnegative real number and every $A_j,B_j$ is a complex $2\times2$ unitary matrix. The list may be empty, durations may be zero, and none of the unitary matrices is required to have determinant one.

QAlgorithms.Papers.Independent.Intakethird100.P99.PureInteraction

For a complex $4\times4$ matrix $H$, indexed by pairs of Boolean values, this predicate holds precisely when there exist nine real coefficients $c_{ij}$, for $i,j\in\{0,1,2\}$, such that

$$H=\sum_{i=0}^{2}\sum_{j=0}^{2}c_{ij}\,\sigma_i\otimes\sigma_j,$$

where $(\sigma_0,\sigma_1,\sigma_2)=(X,Y,Z)$ are the usual Pauli matrices and $\otimes$ is the Kronecker product. The coefficients are unrestricted real numbers, including zero, so the zero matrix is included.

QAlgorithms.Papers.Independent.Intakethird100.P99.SMajorized

For arbitrary real triples $b$ and $a$, this predicate holds precisely when the following three inequalities hold. For each $u\in\{a,b\}$, put $h_u=\max_i|u_i|$, $\ell_u=\min_i|u_i|$, $m_u=|u_0|+|u_1|+|u_2|-h_u-\ell_u$, and $s_u=-\ell_u$ if $u_0u_1u_2<0$, otherwise $s_u=\ell_u$. The required inequalities are

$$h_b\le h_a,\qquad h_b+m_b-s_b\le h_a+m_a-s_a,\qquad h_b+m_b+s_b\le h_a+m_a+s_a.$$

No ordering or nonzero-coordinate assumptions are imposed on either input triple.

QAlgorithms.Papers.Independent.Intakethird100.P99.SOrdered

For a real triple $a=(a_0,a_1,a_2)$, this predicate means precisely that $a_1\le a_0$ and $|a_2|\le a_1$. Both inequalities allow equality, including the zero triple.

QAlgorithms.Papers.Independent.Intakethird100.P99.SamePhysicalGate

For any two complex $4\times4$ matrices $U,V$, indexed by pairs of Boolean values, this predicate holds precisely when there exists a complex scalar $z$ with $|z|=1$ such that

$$U=zV.$$

Neither matrix is assumed unitary; in particular, both may be zero.

QAlgorithms.Papers.Independent.Intakethird100.P99.TransmissionProfile

For every complex matrix $U$ on $\mathbb C^2\otimes\mathbb C^2$ and every $k\in\{0,1,2,3\}$, define $U^{\mathrm{rev}}=SUS$, where $S(x\otimes y)=y\otimes x$; use the standard basis vectors $e_0,e_1$, with $0=\mathrm{false}$ and $1=\mathrm{true}$, and tensor coordinates $(x\otimes y)_{ij}=x_i y_j$. To state the predicate explicitly, let $C(B)$ mean that $B$ is unitary and there exist single-qubit unitary matrices $E,D$, a unit vector $v$, and unit vectors $w_i$ for $i\in\{0,1\}$, such that $B(Ee_i\otimes v)=w_i\otimes De_i$ for every $i$. Let $D_2(B)$ mean that $B$ is unitary and there exist single-qubit unitary matrices $E,F$ and unit vectors $a_{ij},b_{ij}$ for every $i,j\in\{0,1\}$, such that $\langle a_{i0},a_{i1}\rangle=0$ for every $i$, $\langle b_{0j},b_{1j}\rangle=0$ for every $j$, and $B(Ee_i\otimes Fe_j)=a_{ij}\otimes b_{ij}$ for every $i,j$. Let $Q(B)$ mean that $B$ is unitary and there exist single-qubit unitary matrices $E,R$ and unit vectors $v,w$ such that $(I_2\otimes R)B(Eq\otimes v)=w\otimes q$ for every unit vector $q\in\mathbb C^2$. Let $H(B)$ mean that $B$ is unitary and there exist single-qubit unitary matrices $E,F$, single-qubit unitary matrices $R_j$ for $j\in\{0,1\}$, and orthogonal unit vectors $a_0,a_1$, such that $(I_2\otimes R_j)B(Eq\otimes Fe_j)=a_j\otimes q$ for every $j$ and every unit vector $q$. Finally, let $Q_2(B)$ mean that $B$ is unitary and there exist single-qubit unitary matrices $E,F,R,T$ such that $(R\otimes T)B(Eq\otimes Fr)=r\otimes q$ for every pair of unit vectors $q,r\in\mathbb C^2$. The defined profile requires exactly the following truth values, where a false entry means the negation of the corresponding existential predicate:

$$\begin{array}{c|cccccccc} k&C(U)&C(U^{\mathrm{rev}})&D_2(U)&Q(U)&Q(U^{\mathrm{rev}})&H(U)&H(U^{\mathrm{rev}})&Q_2(U)\\ \hline 0&\mathrm F&\mathrm F&\mathrm F&\mathrm F&\mathrm F&\mathrm F&\mathrm F&\mathrm F\\ 1&\mathrm T&\mathrm T&\mathrm F&\mathrm F&\mathrm F&\mathrm F&\mathrm F&\mathrm F\\ 2&\mathrm T&\mathrm T&\mathrm T&\mathrm T&\mathrm T&\mathrm T&\mathrm T&\mathrm F\\ 3&\mathrm T&\mathrm T&\mathrm T&\mathrm T&\mathrm T&\mathrm T&\mathrm T&\mathrm T \end{array}$$

There is no separate unitarity assumption on $U$; in particular, every nonunitary $U$ satisfies the $k=0$ branch.

QAlgorithms.Papers.Independent.Intakethird100.P99.Triple

This is the type of real triples $a=(a_0,a_1,a_2)$, with no restrictions on their entries.

QAlgorithms.Papers.Independent.Intakethird100.P99.TwoMatrix

This is the type of all complex matrices whose rows and columns are indexed by ordered pairs of Boolean values, hence all $4\times4$ complex matrices, with no further restrictions.

QAlgorithms.Papers.Independent.Intakethird100.P99.TwoQubitLocalEquivalent

For arbitrary $4\times4$ complex matrices $U$ and $V$ indexed by Boolean pairs, this predicate means that there exist four unitary $2\times2$ complex matrices $A,B,C,D$ such that $U=(A\otimes B)V(C\otimes D)$. Neither $U$ nor $V$ is separately required to be unitary, and the equality is exact.

QAlgorithms.Papers.Independent.Intakethird100.P99.branchCost

For arbitrary real triples $a=(a_0,a_1,a_2)$ and $b=(b_0,b_1,b_2)$, define, for each real triple $v$,

$$h_v=\max(|v_0|,|v_1|,|v_2|),\quad \ell_v=\min(|v_0|,|v_1|,|v_2|),\quad s(v)=\left(h_v,\ |v_0|+|v_1|+|v_2|-h_v-\ell_v,\ \begin{cases}-\ell_v&v_0v_1v_2<0,\\ \ell_v&v_0v_1v_2\ge0.\end{cases}\right).$$

The branch cost is the infimum in $[0,\infty]$ of the nonnegative real numbers $t$ for which, writing $x=s(b)$ and $y=s(ta)$ with $(ta)_i=ta_i$,

$$x_0\le y_0,\qquad x_0+x_1-x_2\le y_0+y_1-y_2,\qquad x_0+x_1+x_2\le y_0+y_1+y_2.$$

No ordering or nonzero assumptions are imposed on either triple, and $t=0$ is included among the candidate times. An empty set of candidate times has infimum $+\infty$; an infimum is not required to be attained.

QAlgorithms.Papers.Independent.Intakethird100.P99.canonicalPauliInteraction

For every real triple $a=(a_0,a_1,a_2)$, this defines the $4\times4$ complex matrix $a_0(X\otimes X)+a_1(Y\otimes Y)+a_2(Z\otimes Z)$, where, in Boolean order $(\mathrm{false},\mathrm{true})$, $X=\begin{pmatrix}0&1\\1&0\end{pmatrix}$, $Y=\begin{pmatrix}0&-\mathrm i\\ \mathrm i&0\end{pmatrix}$, and $Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$.

QAlgorithms.Papers.Independent.Intakethird100.P99.contentGate

For every real triple $b=(b_0,b_1,b_2)$, this defines the matrix exponential $\exp\!\bigl(-\mathrm i[b_0(X\otimes X)+b_1(Y\otimes Y)+b_2(Z\otimes Z)]\bigr)$, where $X=\begin{pmatrix}0&1\\1&0\end{pmatrix}$, $Y=\begin{pmatrix}0&-\mathrm i\\ \mathrm i&0\end{pmatrix}$, and $Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ in Boolean order $(\mathrm{false},\mathrm{true})$. No ordering or size restrictions are imposed on $b$.

QAlgorithms.Papers.Independent.Intakethird100.P99.contentIntegerShift

For any real triple $b=(b_0,b_1,b_2)$ and any integer triple $n=(n_0,n_1,n_2)$, this function returns the real triple whose coordinates are

$$b_i+\frac{\pi}{2}n_i,\qquad i=0,1,2.$$

Each integer coordinate may be positive, zero, or negative; no reduction modulo any period is performed.

QAlgorithms.Papers.Independent.Intakethird100.P99.controlledContent

For every real number $b$, this defines the real triple

$$\left(\min\left\{b,\frac{\pi}{2}-b\right\},0,0\right).$$

There is no restriction on $b$; in particular, the first coordinate can be negative when $b$ lies outside $[0,\pi/2]$.

QAlgorithms.Papers.Independent.Intakethird100.P99.dcnot

This defines a complex matrix $D$ whose rows and columns are indexed by pairs of Booleans: for every row $p=(p_1,p_2)$ and column $q=(q_1,q_2)$,

$$D_{p,q}= \begin{cases} 1,&p=(q_2,q_1\mathbin{\oplus}q_2),\\ 0,&\text{otherwise}, \end{cases}$$

where $\oplus$ is exclusive-or. Equivalently, on coordinate basis vectors it sends $|q_1,q_2\rangle$ to $|q_2,q_1\oplus q_2\rangle$.

QAlgorithms.Papers.Independent.Intakethird100.P99.interactionCost

For arbitrary complex matrices $H,U$ acting on $\mathbb C^{\{\mathrm{false},\mathrm{true}\}^2}$, the interaction cost is the infimum, in $[0,\infty]$, of all nonnegative real times $t$ satisfying the following condition: $H$ is Hermitian, $U$ is unitary, and for every $\varepsilon>0$ there exist a finite list of nonnegative durations $t_1,\ldots,t_m$, single-qubit unitaries $A_0,B_0,\ldots,A_m,B_m$, and a complex number $z$ with $|z|=1$, such that

$$\sum_{j=1}^m t_j=t,\qquad \left\|(A_m\otimes B_m)e^{-it_mH}\cdots (A_1\otimes B_1)e^{-it_1H}(A_0\otimes B_0)-zU\right\|_{\mathrm{op}}<\varepsilon.$$

Here the norm is the operator norm for the Euclidean norm, and the tensor products are ordinary Kronecker products. The list may be empty, in which case its total duration is zero and its matrix is $A_0\otimes B_0$. The protocols and $z$ may depend on $\varepsilon$. The infimum need not be attained; if there are no admissible times, the cost is $+\infty$, including whenever $H$ is not Hermitian or $U$ is not unitary.

QAlgorithms.Papers.Independent.Intakethird100.P99.interactionPhases

For every real triple $b=(b_0,b_1,b_2)$, this defines a function on $\{0,1,2,3\}$ with values, in that order,

$$\left( b_0+b_1-b_2,\quad b_0-b_1+b_2,\quad -b_0+b_1+b_2,\quad -b_0-b_1-b_2 \right).$$

There are no assumptions on $b$, and no reduction of these real values modulo a period is performed.

QAlgorithms.Papers.Independent.Intakethird100.P99.localMatrix

For any unitary $2\times2$ complex matrices $A$ and $B$ indexed by Boolean values, this defines their Kronecker product $A\otimes B$, whose entry at row $(x,y)$ and column $(x',y')$ is $A_{x,x'}B_{y,y'}$.

QAlgorithms.Papers.Independent.Intakethird100.P99.magicBasis

This defines the complex matrix with rows indexed by Boolean pairs and columns indexed by $0,1,2,3$ which, in row order $(\mathrm{false},\mathrm{false}),(\mathrm{false},\mathrm{true}),(\mathrm{true},\mathrm{false}),(\mathrm{true},\mathrm{true})$, is

$$B=\frac{1}{\sqrt2} \begin{pmatrix} 0&1&-i&0\\ -i&0&0&1\\ -i&0&0&-1\\ 0&1&i&0 \end{pmatrix}.$$

QAlgorithms.Papers.Independent.Intakethird100.P99.magicMatrix

For every complex matrix $U$ whose rows and columns are indexed by Boolean pairs, this defines the complex matrix on indices $0,1,2,3$ by

$$B^\dagger U B, \qquad B=\frac{1}{\sqrt2} \begin{pmatrix} 0&1&-i&0\\ -i&0&0&1\\ -i&0&0&-1\\ 0&1&i&0 \end{pmatrix},$$

where the rows of $B$ are ordered as $(\mathrm{false},\mathrm{false}),(\mathrm{false},\mathrm{true}),(\mathrm{true},\mathrm{false}),(\mathrm{true},\mathrm{true})$, its columns are ordered $0,1,2,3$, and $B^\dagger$ is its conjugate transpose. No unitarity assumption on $U$ is required.

QAlgorithms.Papers.Independent.Intakethird100.P99.optimalTransmissionContent

For every real triple $a=(a_0,a_1,a_2)$, this defines the real triple

$$\left(\frac{\pi}{4},\frac{\pi}{4}, \frac{\pi}{4}\frac{2a_2}{a_0+a_1}\right).$$

There are no ordering or nonzero-denominator assumptions on $a$. Division is total: if $a_0+a_1=0$, the third coordinate is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P99.protocolMatrix

For any complex $4\times4$ matrix $H$, indexed by pairs of Boolean values, and any pulse protocol with initial complex $2\times2$ unitaries $A_0,B_0$ and ordered list $(t_j,A_j,B_j)_{j=1}^{m}$, where $t_j\ge0$ and $A_j,B_j$ are complex $2\times2$ unitaries, define

$$V_0=A_0\otimes B_0,\qquad V_j=(A_j\otimes B_j)\exp(-\mathrm{i}t_jH)V_{j-1}\quad(1\le j\le m).$$

The protocol matrix is $V_m$. For an empty list it is $A_0\otimes B_0$; zero durations are allowed, and $H$ is not assumed Hermitian.

QAlgorithms.Papers.Independent.Intakethird100.P99.protocolTime

For any pulse protocol $p$, consisting of initial complex $2\times2$ unitaries and a finite ordered list $(t_j,A_j,B_j)_{j=1}^{m}$ of nonnegative real durations paired with two complex $2\times2$ unitaries, its protocol time is the nonnegative real number

$$\sum_{j=1}^{m}t_j.$$

The initial unitary matrices do not contribute to this sum, and an empty list has protocol time zero.

QAlgorithms.Papers.Independent.Intakethird100.P99.reverseParties

For every complex matrix $U$ indexed by Boolean pairs, this defines $SUS$, where $S$ is the matrix with entry $S_{p,q}=1$ when $p=(q_2,q_1)$ and entry zero otherwise. Thus the resulting entries are

$$\bigl(\operatorname{reverseParties}(U)\bigr)_{(a,b),(c,d)} =U_{(b,a),(d,c)}$$

for all Booleans $a,b,c,d$. There is no unitarity assumption on $U$.

QAlgorithms.Papers.Independent.Intakethird100.P99.sOrder

For any real triple $a=(a_0,a_1,a_2)$, define $h=\max\{|a_0|,|a_1|,|a_2|\}$ and $\ell=\min\{|a_0|,|a_1|,|a_2|\}$. The function returns the real triple

$$\left(h,\ |a_0|+|a_1|+|a_2|-h-\ell,\ \begin{cases} -\ell,&a_0a_1a_2<0,\\ \ell,&a_0a_1a_2\ge0. \end{cases}\right).$$

Thus the first two coordinates are the largest and middle absolute values; the third is the smallest absolute value with the indicated sign. Ties and zero coordinates are included, and a zero product takes the nonnegative branch.

QAlgorithms.Papers.Independent.Intakethird100.P99.secondShift

For any real triple $b=(b_0,b_1,b_2)$, this function returns

$$\left(b_0-\frac{\pi}{2},\,b_1,\,b_2\right).$$

There are no ordering or range restrictions on $b$.

QAlgorithms.Papers.Independent.Intakethird100.P99.taskCost

For an arbitrary complex two-qubit matrix $H$ and an arbitrary predicate $P$ on complex two-qubit matrices, this defines $\inf\{C_H(U):P(U)\}$ in $[0,\infty]$. Here $C_H(U)$ is the infimum of the nonnegative real times $t$ for which $H$ is Hermitian, $U$ is unitary, and for every $\varepsilon>0$ there exist nonnegative durations $t_1,\ldots,t_m$ summing exactly to $t$, single-qubit unitaries $A_0,B_0,\ldots,A_m,B_m$, and a complex number $z$ with $|z|=1$, such that

$$\left\|(A_m\otimes B_m)e^{-it_mH}\cdots (A_1\otimes B_1)e^{-it_1H}(A_0\otimes B_0)-zU\right\|_{\mathrm{op}}<\varepsilon.$$

The norm is the Euclidean operator norm; the list is finite and may be empty, yielding duration zero and matrix $A_0\otimes B_0$. The protocol and phase may depend on $\varepsilon$. Both infima use the convention $\inf\varnothing=+\infty$, and neither infimum is required to be attained. There is no separate restriction on $H$ or on the matrices satisfying $P$.