Quantum t-design

A Quantum t-design is a probability distribution over pure quantum states which can duplicate properties of the probability distribution over the Haar measure for polynomials of degree t or less. Specifically, the average of any polynomial function of degree t over the design is exactly the same as the average over Haar measure. Here the Haar measure is a uniform probability distribution over all quantum states. Quantum t-designs are so called because they are analogous to t-designs in classical statistics, which arose historically in connection with the problem of design of experiments. Quantum t-designs are usually unique, and thus almost always calculable. Two particularly important types of t-designs in quantum mechanics are spherical and unitary t-designs.

Spherical t-designs are designs where points of the design (i.e. the points being used for the averaging process) are points on a unit sphere. Spherical t-designs and variations thereof have been considered lately and found useful in quantum information theory,^[1] quantum cryptography and other related fields.

Unitary designs are analogous to spherical designs in that they approximate the entire unitary group via a finite collection of unitary matrices. Unitary designs have been found useful in information theory^[2] and quantum computing. Unitary designs are especially useful in quantum computing since most operations are represented by unitary operators.

Motivation

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 $\tfrac{1}{2}$ 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.

Another recent application is the fact that a symmetric informationally complete POVM is also a spherical 2-design. Also, since a 2-design must have more than $d^2$ elements, a SIC-POVM is a minimal 2-design.

Spherical Designs

Complex projective (t,t)-designs have been studied in quantum information theory as quantum 2-designs, and in t-designs of vectors in the unit sphere in $\mathbb{R}^N$ which, when transformed to vectors in $\mathbb{C}^{N/2}$ become complex projective (t/2,t/2)-designs.

Formally, we define^[3] a complex projective (t,t)-design as a probability distribution over quantum states $(p_i,|\phi_i\rangle)$ if

$\sum_i p_i (|\phi_i\rangle \langle \phi_i|)^{\otimes t} = \int_{\psi}(|\psi\rangle \langle \psi|)^{\otimes t}d\psi$

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

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

The technical definition of an approximate (t,t)-design is:

If $\sum_i p_i |\phi_i\rangle \langle \phi_i| = \int_{\psi}|\psi\rangle \langle \psi|d\psi$

and $(1-\epsilon)\int_{\psi}(|\psi\rangle \langle \psi|)^{\otimes t}d\psi \leq \sum_i p_i (|\phi_i\rangle \langle \phi_i|)^{\otimes t} \leq (1+\epsilon)\int_{\psi}(|\psi\rangle \langle \psi|)^{\otimes t}d\psi$

then $(p_i,|\phi_i\rangle)$ is an $\epsilon$-approximate (t,t)-design.

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

Construction

For convenience N is assumed to be a power of 2.

Using the fact that for any N there exists a set of $N^d$ functions {0,...,N-1} $\rightarrow$ {0,...,N-1} such that for any distinct $k_1, ..., k_d \in$ {0,...,N-1} the image under f, where f is chosen at random from S, is exactly the uniform distribution over tuples of d elements of {0,...,N-1}.

Let $|\psi\rangle = \sum_{i=1}^N \alpha |i\rangle$ be drawn from the Haar measure. Let $P_n$ be the probability distribution of $\alpha_n$ and let $P= \lim_{N\rightarrow \infty} \sqrt{N} P_N$. Finally let $\alpha$ be drawn from P. If we define $X = |\alpha|$ with probability $\tfrac12$ and $X = -|\alpha|$ with probability $\tfrac{1}{2}$ then: $E[X^j] = 0$ for odd j and $E[X^j] = (\tfrac{j}{2})!$ for even j.

Using this and Gaussian quadrature we can construct $p_{f,g} = \frac{\sum_{i=1}^N a_{f,i}^2}{|S_1| |S_2|}$ so that $p_{f,g}|\psi_{f,g}\rangle$ is an approximate (t,t)-design.

Unitary Designs

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

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

Formally define^[4] a unitary t-design, X, if

$\frac{1}{|X|}\sum_{U \in X} U^{\otimes t}\otimes (U^{*})^{\otimes t} = \int_{U(d)} U^{\otimes t}\otimes (U^{*})^{\otimes t}dU$

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

Using the permutation maps it is possible^[3] to verify directly that a set of unitary matrices forms a t-design.^[5]

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

$\frac{1}{|X|^2} \sum_{U,V \in X}|tr(U*V)|^{2t} \geq \int_{U(d)}|tr(U*V)|^{2t}dU$

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.^[6]

Bounds for unitary designs

Define $Hom(U(d),t,t)$ as the set of functions homogeneous of degree t in $U$ and homogeneous of degree t in $U^{*}$, then if for every $f \in Hom(U(d),t,t)$:

$\frac{1}{|X|} \sum_{U \in X} f(U) = \int_{U(d)}f(U) dU$

then X is a unitary t-design.

We further define the inner product for functions $f$ and $g$ on $U(d)$ as the average value of $\bar{f}g$ as:

$\langle f,g\rangle := \int_{U(d)}\bar{f(U)}g(U) dX$

and $\langle f,g\rangle_X$ as the average value of $\bar{f}g$ over any finite subset $X \subset U(d)$.

it follows that X is a unitary t-design iff $\langle 1,f\rangle_X = \langle 1,f\rangle \quad\forall f$.

From the above it is demonstrable that if X is a t-design then $|X| \geq dim(Hom(U(d),\left\lceil\tfrac{t}2\right\rceil,\left\lfloor\tfrac{t}2\right\rfloor))$ is an absolute bound for the design. This imposes an upper bound on the size of a unitary design. This bound is absolute meaning it depends only on the strength of the design or the degree of the code, and not the distances in the subset, X.

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A unitary code is a finite subset of the unitary group in which a few inner product values occur between elements. Specifically, a unitary code is defined as a finite subset $X \subset U(d)$ if for all $U \neq M$ in X $|tr(U^*M)|^2$ takes only distinct values.

It follows that $|X| \leq dim(Hom(U(d),s,s))$ and if U and M are orthogonal: $|X| \leq dim(Hom(U(d),s,s-1))$

Notes

1. A. Hayashi, T. Hashimoto, M. Horibe. Reexamination of optimal quantum state estimation of pure states. Phys. Rev. A, 72: 032325, 2006. Also quant-ph/0410207.
2. C. Dankert, R. Cleve, J. Emerson, and E. Livine, Exact and approximate unitary 2-designs: constructions and applications, (2006).
3. [quant-ph/0701126] Quantum t-designs: t-wise independence in the quantum world
4. [0809.3813] Unitary designs and codes
5. B. Collins and P. ´ Sniady, Integration with respect to the Haar measure on unitary, orthogonal and symplectic group, Comm. Math. Phys.,264 (2006), 773–795.
6. D. Gross, K. Audenaert, and J. Eisert, Evenly distributed unitaries: on the structure of unitary designs, J. Math. Phys., 48 (2007),052104, 22.