Discrete Quantum Walks and Szegedy Theory

A finite row-stochastic kernel is a finite set $X$ and a matrix $P=(P_{xy}{)}_{x,y\in X}$ with $P_{xy}\in [0,1]$ and

${\sum }_{y\in X}{P}_{xy}=1(x\in X)\mathrm{.}$

The row convention is $P_{xy}=\mathrm{Pr}(Y=y\mid X=x)$: distributions are row vectors and one transition sends $\mu$ to $\mu P$. If $X=\mathrm{\varnothing }$, the row conditions are vacuous and there is no probability distribution on $X$. No stationary distribution, reversibility, irreducibility, aperiodicity, or efficient circuit access is part of this definition.

A finite reversible Markov chain is a finite row-stochastic kernel $(X,P)$ together with a supplied probability vector $\pi =({\pi }_x{)}_{x\in X}$ such that ${\pi }_x>0$ for every $x$, ${\sum }_x{\pi }_x=1$, $\pi P=\pi$, and

${\pi }_x{P}_{xy}={\pi }_y{P}_{yx}(x,y\in X)\mathrm{.}$

The stationary law and detailed balance are explicit input hypotheses; the definition does not invoke a theorem that produces a stationary law. No irreducibility or aperiodicity is assumed. In particular, a periodic reversible chain is included.

For a finite row-stochastic kernel $(X,P)$, let ${\mathbb{C}}^X$ and ${\mathbb{C}}^{X\times X}$ carry the standard orthonormal label and ordered-product bases. If $X\mathrm{\ne }\mathrm{\varnothing }$, these coordinate spaces are implemented as two ordered registers; if $X=\mathrm{\varnothing }$, they are zero spaces and every map below is the unique zero-dimensional map. Define the row-preparation linear map $T:{\mathbb{C}}^X\to {\mathbb{C}}^{X\times X}$ and the swap $S$ on basis vectors by

$T\mathrm{\mid }x⟩=\mathrm{\mid }x⟩\otimes {\sum }_{y\in X}\sqrt{P_{xy}}\mathrm{\mid }y⟩,S\mathrm{\mid }x,y⟩=\mathrm{\mid }y,x⟩\mathrm{.}$

Every scalar square root is the unique nonnegative real root. Define the discriminant by $D:=T^{\ast }ST$, using the adjoint characterized by $⟨Tv,w⟩=⟨v,T^{\ast }w⟩$ for the root inner product, which is linear in its first argument. The row-preparation map is an isometry; its complete verification is The Szegedy discriminant is self-adjoint and contractive ↗. This is the general row-stochastic construction and assumes no reversibility.

Let $(X,P,\pi )$ be a finite reversible chain and let $f:X\to \mathbb{C}$. Define $\mathrm{\parallel }f{\mathrm{\parallel }}_{2,\pi }^2:={\sum }_x{\pi }_x\mathrm{\mid }{f}_x{\mathrm{\mid }}^2$ and $(Pf{)}_x:={\sum }_y{P}_{xy}{f}_y$. Then

$\mathrm{\parallel }Pf{\mathrm{\parallel }}_{2,\pi }^2\le \mathrm{\parallel }f{\mathrm{\parallel }}_{2,\pi }^2,$

so $P$ is a contraction on the finite weighted inner-product space $L^2(X,\pi )$. Detailed balance is not needed for this inequality beyond the stated stationary-law hypotheses.

For a finite reversible chain $(X,P,\pi )$, equip ${\mathbb{C}}^X$ with $⟨f,g{⟩}_{\pi }:={\sum }_x{\pi }_x{f}_x\overline{g_x}$. Then $P$ is self-adjoint for this inner product, and the mean-zero subspace

$H_0:=\{f:{\sum }_x{\pi }_x{f}_x=0\}=\{f:⟨f,\mathbf{1}{⟩}_{\pi }=0\}$

is invariant under $P$. The restriction $P{\mathrm{\mid }}_{H_0}$ is therefore a well-defined endomorphism and is self-adjoint for the restricted inner product.

For a finite reversible chain $(X,P,\pi )$, let $H_0=\{f:{\sum }_x{\pi }_x{f}_x=0\}$ with the weighted inner product from A finite reversible transition operator is self-adjoint on weighted L2. Define its absolute spectral gap by

${\delta }_{\mathrm{a}\mathrm{b}\mathrm{s}}(P):=1-\mathrm{\parallel }P{\mathrm{\mid }}_{H_0}{\mathrm{\parallel }}_{2,\pi }\mathrm{.}$

The norm of the restriction to the zero subspace is $0$ by The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||, so a one-state chain has absolute gap $1$. No irreducibility or aperiodicity is built into this definition: a reducible chain with a nonconstant stationary vector in $H_0$ can have gap $0$, and a periodic chain can have gap $0$ because of an eigenvalue $-1$. The eigenvalue form of the definition is proved in Absolute gap equals one minus the largest nonconstant eigenvalue modulus.

Let $(X,P,\pi )$ be a finite reversible chain and let $H_0=\{f:{\sum }_x{\pi }_x{f}_x=0\}$. If $H_0=\{0\}$, then ${\delta }_{\mathrm{a}\mathrm{b}\mathrm{s}}(P)=1$. Otherwise,

${\delta }_{\mathrm{a}\mathrm{b}\mathrm{s}}(P)=1-\max \{\mathrm{\mid }\lambda \mathrm{\mid }:\lambda \in \sigma (P{\mathrm{\mid }}_{H_0})\},$

and $0\le {\delta }_{\mathrm{a}\mathrm{b}\mathrm{s}}(P)\le 1$. If there is a fixed function $f\mathrm{\notin }\mathrm{span}\{\mathbf{1}\}$, then its nonzero mean-zero component is fixed by $P$, so the gap is $0$.

For every finite row-stochastic kernel, the map $T$ of Szegedy row-preparation map and discriminant satisfies $T^{\ast }T=I$ and is an isometry, and the matrix entries of $D=T^{\ast }ST$ are

$D_{xy}=\sqrt{P_{xy}{P}_{yx}}\mathrm{.}$

Hence $D$ is real symmetric, self-adjoint, and $\mathrm{\parallel }D\mathrm{\parallel }\le 1$. If $(X,P,\pi )$ is also finite reversible, then

$D=\mathrm{diag}(\sqrt{\pi })P\mathrm{diag}(1\mathrm{/}\sqrt{\pi })\mathrm{.}$

Every eigenvalue of $D$ is real and belongs to $[-1,1]$. The empty state set has the zero endomorphism and empty spectrum; the similarity assertion is only made when a positive stationary probability vector is supplied.

For the finite row-stochastic kernel and row-preparation map of Szegedy row-preparation map and discriminant, let $\mathrm{\Pi }:=T{T}^{\ast }$, the orthogonal projection onto $\mathrm{ran}(T)$, and choose the reflection sign

$R_A:=2\mathrm{\Pi }-I\mathrm{.}$

With $S$ the two-register swap, define the one-reflection Szegedy step and its square by

$U:=S{R}_A,W:=U^2=(S{R}_{A}S)R_A=(2S\mathrm{\Pi }S-I)(2\mathrm{\Pi }-I)\mathrm{.}$

Products act right to left. This fixes $U$ and $W$ as different operators. Algebraically, if $Uv=\mu v$, then $Wv={\mu }^{2}v$ on the same vector. The other common reflection sign is $I-2\mathrm{\Pi }=-R_A$ and is not interchangeable inside a controlled operation.

For the projection $\mathrm{\Pi }=T{T}^{\ast }$ onto $A=\mathrm{ran}(T)$, the operators $R_A=2\mathrm{\Pi }-I$ and $S$ are self-adjoint involutions. Hence $U=S{R}_A$ and $W=U^2$ are unitary. On $A^{\perp }$, $\mathrm{\Pi }=0$ and therefore $U=-S$.

For any eigenpair $Dv=\lambda v$ with $v\mathrm{\ne }0$ and $\lambda \in [-1,1]$, put $a:=Tv$ and $b:=STv$. Then $\mathrm{span}\{a,b\}$ is invariant under $U$, and

$Ua=b,Ub=-a+2\lambda b\mathrm{.}$

All eigenvalues of $D$ satisfy the displayed interval hypothesis by The Szegedy discriminant is self-adjoint and contractive. If $\lambda =1$ or $\lambda =-1$, then $b=\lambda a$ and the span is one-dimensional. If $-1<\lambda <1$, the two vectors are linearly independent and the invariant subspace is two-dimensional.

For a nonempty finite row-stochastic kernel $(X,P)$, a row-preparation access model supplies an explicit bijection $\eta :X\to \{0,\dots ,\mathrm{\mid }X\mathrm{\mid }-1\}$; write ${\kappa }_x$ for the image of $x$ under this bijection. Every ket $\mathrm{\mid }x⟩$ below denotes the basis vector $\mathrm{\mid }{\kappa }_x⟩$. The model supplies a finite work register $W$ with a specified clean unit vector $\mathrm{\mid }{0}_W⟩$ and a unitary $V_P$ on the ordered register product ${\mathbb{C}}^X\otimes {\mathbb{C}}^X\otimes W$ satisfying

$V_P(\mathrm{\mid }x⟩\otimes \mathrm{\mid }0⟩\otimes \mathrm{\mid }{0}_W⟩)=(\mathrm{\mid }x⟩\otimes {\sum }_y\sqrt{P_{xy}}\mathrm{\mid }y⟩)\otimes \mathrm{\mid }{0}_W⟩(x\in X)\mathrm{.}$

The state encoding and its description are part of the supplied circuit interface and are included in the circuit description or uniform generator. The model separately declares access and costs for the forward call $V_P$, inverse call $V_P^{\ast }$, controlled $V_P$, controlled $V_P^{\ast }$, a direct range-reflection call, controlled range-reflection access, the clean-state reflection $R_0$, controlled $R_0$, the swap on the first two registers, and controlled swap access. The clean-state reflection is

$R_0:=2(I_X\otimes \mathrm{\mid }0,0_W⟩⟨0,0_W\mathrm{\mid })-I,$

where the second register and work register are in their specified clean states. When range reflection is implemented from row preparation, the supplied circuit word is $V_P{R}_0{V}_P^{\ast }$; the access-cost theorem proves that on the clean-work subspace it realizes the logical reflection $2T{T}^{\ast }-I$. The swap acts as $S$ on the first two registers and as the identity on $W$.

Query calls, gate counts, circuit-description bits, uniform-generation time, and precision are separate cost coordinates. A controlled walk implemented factor by factor needs controlled access to each constituent, including the preparation calls, $R_0$, and the swap. A matrix $P$ by itself supplies none of these circuits or efficient costs. For a family of encoded chains, a uniform circuit family is an additional explicit input, not a consequence of the finite matrix definition.

Let $D=T^{\ast }ST$ be the discriminant of a finite row-stochastic kernel and let $U=S(2T{T}^{\ast }-I)$ and $W=U^2$. For every eigenvalue $\lambda \in (-1,1)$ of $D$, counted with multiplicity, its invariant planes contribute eigenvalues

$\exp (i\arccos \lambda ),\exp (-i\arccos \lambda )\text{for }U,\exp (2i\arccos \lambda ),\exp (-2i\arccos \lambda )\text{for }W\mathrm{.}$

For each eigendirection with $\lambda =1$ (respectively $\lambda =-1$), the plane collapses to a line on which $U$ acts as $+I$ (respectively $-I$), so $W$ acts as $+I$. On the remaining subspace $K=(\mathrm{ran}T+\mathrm{ran}(ST){)}^{\perp }$, $U=-S$ and its eigenvalues lie in $\{+1,-1\}$; therefore all eigenvalues of $W$ there are $+1$. If $X=\mathrm{\varnothing }$, both spaces are zero and the statements about eigenvalues are vacuous.

For a nonempty chain in the row-preparation implementation of Finite row-preparation access model for a Szegedy walk, the full-workspace word $S{V}_P{R}_0{V}_P^{\ast }$ restricts on the clean-work subspace to the logical one-reflection step $U=S(2T{T}^{\ast }-I)$. One such uncontrolled step uses exactly one call each to $V_P^{\ast }$ and $V_P$, one clean-state reflection $R_0$, and one swap $S$. The factorwise controlled implementation controls each constituent, including both row-preparation calls, $R_0$, and $S$; this count describes that implementation. A $W=U^2$ step uses exactly two such $U$ implementations. Query calls, gate counts, description bits, uniform-generation time, and precision are recorded as separate cost coordinates.