Quantum t-design

In a d-dimensional Hilbert space when averaging over all quantum pure states the natural group is SU(d), the special unitary group of dimension d. The Haar measure is, by definition, the unique group-invariant measure, so it is used to average properties that are not unitarily invariant over all states, or over all unitaries.

A particularly widely used example of this is the spin ${\displaystyle {\tfrac {1}{2}}}$ system. For this system the relevant group is SU(2) which is the group of all 2x2 unitary operators. Since every 2x2 unitary operator is a rotation of the Bloch sphere, the Haar measure for spin-1/2 particles is invariant under all rotations of the Bloch sphere. This implies that the Haar measure is the rotationally invariant measure on the Bloch sphere, which can be thought of as a constant density distribution over the surface of the sphere.

An important class of complex projective t-designs, are symmetric informationally complete positive operator-valued measures POVM's, which are complex projective 2-design. Since such 2-designs must have at least ${\displaystyle d^{2}}$ elements, a SIC-POVM is a minimal sized complex projective 2-designs.

Complex projective t-designs have been studied in quantum information theory as quantum t-designs. These are closely related to spherical 2t-designs of vectors in the unit sphere in ${\displaystyle \mathbb {R} ^{d}}$ which when naturally embedded in ${\displaystyle \mathbb {C} ^{d}}$ give rise to complex projective t-designs.

Formally, we define a probability distribution over quantum states ${\displaystyle (p_{i},|\phi _{i}\rangle )}$ to be a [5] complex projective t-design if

${\displaystyle \sum _{i}p_{i}(|\phi _{i}\rangle \langle \phi _{i}|)^{\otimes t}=\int _{\psi }(|\psi \rangle \langle \psi |)^{\otimes t}d\psi }$

Here, the integral over states is taken over the Haar measure on the unit sphere in ${\displaystyle \mathbb {C} ^{d}}$

Exact t-designs over quantum states cannot be distinguished from the uniform probability distribution over all states when using t copies of a state from the probability distribution. However in practice even t-designs may be difficult to compute. For this reason approximate t-designs are useful.

Approximate t-designs are most useful due to their ability to be efficiently implemented. i.e. it is possible to generate a quantum state ${\displaystyle |\phi \rangle }$ distributed according to the probability distribution ${\displaystyle p_{i}|\phi _{i}\rangle }$ in ${\displaystyle O(\log ^{c}d)}$ time. This efficient construction also implies that the POVM of the operators ${\displaystyle Np_{i}|\phi _{i}\rangle \langle \phi _{i}|}$ can be implemented in ${\displaystyle O(\log ^{c}d)}$ time.

The technical definition of an approximate t-design is:

If ${\displaystyle \sum _{i}p_{i}|\phi _{i}\rangle \langle \phi _{i}|=\int _{\psi }|\psi \rangle \langle \psi |d\psi }$

and ${\displaystyle (1-\epsilon )\int _{\psi }(|\psi \rangle \langle \psi |)^{\otimes t}d\psi \leq \sum _{i}p_{i}(|\phi _{i}\rangle \langle \phi _{i}|)^{\otimes t}\leq (1+\epsilon )\int _{\psi }(|\psi \rangle \langle \psi |)^{\otimes t}d\psi }$

then ${\displaystyle (p_{i},|\phi _{i}\rangle )}$ is an ${\displaystyle \epsilon }$-approximate t-design.

It is possible, though perhaps inefficient, to find an ${\displaystyle \epsilon }$-approximate t-design consisting of quantum pure states for a fixed t.

Unitary t-designs are analogous to spherical designs in that they reproduce the entire unitary group via a finite collection of unitary matrices.[1] The theory of unitary 2-designs was developed in 2006 [1] specifically to achieve a practical means of efficient and scalable randomized benchmarking[3] to assess the errors in quantum computing operations, called gates. Since then unitary t-designs have been found useful in other areas of quantum computing and more broadly in quantum information theory and in fields as far reaching as black hole physics.[4] Unitary t-designs are especially relevant to randomization tasks in quantum computing since ideal operations are usually represented by unitary operators.

Elements of a unitary t-design are elements of the unitary group, U(d), the group of ${\displaystyle d\times d}$ unitary matrices. A t-design of unitary operators will generate a t-design of states.

Suppose ${\displaystyle {U_{k}}}$ is a unitary t-design (i.e. a set of unitary operators). Then for any pure state ${\displaystyle |\psi \rangle }$ let ${\displaystyle |\psi _{k}\rangle =U_{k}|\psi \rangle }$. Then ${\displaystyle {|\psi _{k}\rangle }}$ will always be a t-design for states.

Formally define[6] a unitary t-design, X, if

${\displaystyle {\frac {1}{|X|}}\sum _{U\in X}U^{\otimes t}\otimes (U^{*})^{\otimes t}=\int _{U(d)}U^{\otimes t}\otimes (U^{*})^{\otimes t}dU}$

Observe that the space linearly spanned by the matrices ${\displaystyle U^{\otimes r}\otimes (U^{*})^{\otimes s}dU}$ over all choices of U is identical to the restriction ${\displaystyle U\in X}$ and ${\displaystyle r+s=t}$ This observation leads to a conclusion about the duality between unitary designs and unitary codes.

Using the permutation maps it is possible[5] to verify directly that a set of unitary matrices forms a t-design.[7]

One direct result of this is that for any finite ${\displaystyle X\subseteq U(d)}$

${\displaystyle {\frac {1}{|X|^{2}}}\sum _{U,V\in X}|tr(U*V)|^{2t}\geq \int _{U(d)}|tr(U*V)|^{2t}dU}$

With equality if and only if X is a t-design.

1 and 2-designs have been examined in some detail and absolute bounds for the dimension of X, |X|, have been derived.[8]