bibkey: brylinski2001universal
authors: Jean-Luc Brylinski; Ranee Brylinski
year: 2001
title: Universal quantum gates
doi: null
url: https://arxiv.org/abs/quant-ph/0108062v1
claim: A diagonal two-qudit gate preserves all product states exactly when every phase rectangle vanishes modulo two pi; an imprimitive gate together with all one-qudit gates is exactly universal.
strata_touched: []
license: citation-only
triage: anchor

Mixed phases and exact gate generation

Verified source scope

In arXiv:quant-ph/0108062v1, Definition 1.2 distinguishes dense universality from exact finite-product universality. Theorem 1.3 proves that, when all one-qudit unitary gates are supplied, adding a two-qudit gate gives either form of universality precisely when that gate is imprimitive: it sends some product vector to a nonproduct vector. Proposition 2.1 specializes the product-preservation condition to diagonal gates: all phase rectangles must vanish modulo $2\pi$. These statements and their proofs are available in the cited full text.

Theorem 1.4 also allows a product-preserving gate to be a tensor product followed by SWAP. Consequently, failure to be a tensor product alone is not a general entangling criterion. The diagonal case below has the stronger explicit factorization.

Two-qubit phase rectangle

Let $U_{\varphi }|ab⟩=e^{i{\varphi }_{ab}}|ab⟩$ for $a,b\in \{0,1\}$, and set

$$\alpha ={\varphi }_{10}-{\varphi }_{00},\beta ={\varphi }_{01}-{\varphi }_{00},\Theta ={\varphi }_{11}-{\varphi }_{10}-{\varphi }_{01}+{\varphi }_{00}(\mathrm{mod}2\pi ).$$

The elementary two-qubit specialization of Proposition 2.1 is

$$U_{\varphi }=e^{i{\varphi }_{00}}\left[\mathrm{diag}(1,e^{i\alpha })\otimes \mathrm{diag}(1,e^{i\beta })\right]\exp (i\Theta n_1{n}_2),n=|1⟩⟨1|.$$

Checking the four basis vectors proves the equality. In particular, vanishing $\Theta$ gives local factorization. Conversely, $U_{\varphi }|++⟩$ has coefficient matrix

$$M=\frac{1}{2}\left(\begin{array}{cc}{e}^{i{\varphi }_{00}}& e^{i{\varphi }_{01}}\\ e^{i{\varphi }_{10}}& e^{i{\varphi }_{11}}\end{array}\right),\det M=\frac{e^{i({\varphi }_{01}+{\varphi }_{10})}}{4}(e^{i\Theta }-1).$$

A pure bipartite vector is a product exactly when its coefficient matrix has rank one. Thus $\Theta \not\equiv 0$ gives Schmidt rank two and rules out local factorization. This is an established phase-rectangle argument, not a new entanglement test.

For a declared encoder $U_{\theta ,f}|ab⟩=e^{i\theta f(a,b)}|ab⟩$, its invariant is

$$\Theta =\theta [f(1,1)-f(1,0)-f(0,1)+f(0,0)](\mathrm{mod}2\pi ).$$

The raw real mixed difference need not vanish when the encoded gate is local: $f(a,b)=4ab$ and $\theta =\pi /2$ produce the identity. A discrete second difference also does not require analytic nonsmoothness; the polynomial $f(x,y)=xy$ has mixed difference one on the unit square.

Exact elementary decompositions and scalar scope

Barenco et al., Elementary gates for quantum computation, arXiv:quant-ph/9503016v1, abstract and §§4–8, prove finite-product universality of all one-qubit $U(2)$ gates with controlled-NOT. Lemma 4.1 supplies the single-qubit Euler parametrization. The published article is Physical Review A 52, 3457–3467, DOI 10.1103/PhysRevA.52.3457. This note uses its exact decomposition result, not only a density statement or a finite discrete approximation theorem.

If only all single-qubit $SU(2)$ gates and CNOT are supplied, the same finite decomposition realizes every two-qubit unitary channel: replace each $U(2)$ factor by a scalar times an $SU(2)$ factor and collect the scalars into the unobservable overall phase. This does not supply an arbitrary relative phase between a logical subspace and a separately accessible orthogonal sector. Brylinski §9 explicitly distinguishes determinant restrictions for special one-qudit gates; Theorem 1.3 cannot simply be quoted with its $U(d)$ assumption removed.

For local rotations $e^{-itX}$ and $e^{-itY}$, Euler angles are rescaled by the convention that these exponentials have angle $2t$ on the Bloch sphere. They generate each $SU(2)$ element by three rotations. The inference is about ideal real parameters. Their computation, finite precision, physical implementation and errors require separate contracts.

The fixed controlled-S identity used to extract a local phase already appears in Garion–Cross, equation (12). Its application to the constrained physical pulses and the five-mode endpoint code is in Static compatibility fibers §62. Existing native compilers with a larger phase-addressable menu are described in the Numata–Yamanouchi control note; their control permissions must not be substituted for the fixed-phase menu used there.

Static composition retains coherent information

For a finite circuit $U=U_L\cdots U_1$ on fixed finite-dimensional ports, inserting a complete orthonormal basis between adjacent factors gives

$$⟨x_L|U|x_0⟩=\sum _{x_1,\dots ,x_{L-1}}\prod _{t=1}^L⟨x_t|U_t|x_{t-1}⟩.$$

This is ordinary matrix multiplication written as a static tensor contraction. The ports, order and complex amplitudes remain part of the description. A list of local configuration probabilities is insufficient: $(|000⟩+|101⟩)/\sqrt{2}$ and $(|000⟩-|101⟩)/\sqrt{2}$ are orthogonal but have identical occupancy probabilities and means. Squaring individual amplitudes before contraction loses the interference between paths. The identity gives neither a cheap general contraction algorithm nor an identification of a configuration polytope with physical spacetime.

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