Quantum t-designs: t-wise independence in the quantum world

Andris Ambainis, Joseph Emerson

IQC, University of Waterloo

Random quantum states Several recent results using

random quantum objects: Random quantum states; Random unitary transformations; Random orthonormal bases.

Private quantum channels

Alice wants to send | to Bob, over a channel that may be eavesdropped by Eve.

Alice and Bob share a classical secret key i, which they can use to encrypt |.

A B| Eve

Private quantum channels

[Hayden et al., 2001]: Let N = dim |. Let U1, U2, … be O(N log N) unitaries,

known to both Alice and Bob. Alice randomly chooses Ui, sends Ui|.

A B| Ui| |

Private quantum channels

[Hayden et al., 2001]: If U1, U2, …are uniformly random

unitary transformations, Eve gets almost no information about |.

A B| Ui| |

Summary

•Random quantum objects are useful!

•How do we generate and describe a random state?

•A random state on n qubits has 2n amplitudes.

•Since amplitudes are random, 2n are bits required to describe the state.

•Protocols are highly inefficient!

Quantum pseudorandomness We want small sets of quantum

states, with properties similar to random states.

In this talk: quantum counterpart of t-wise independence.

Outline

1. Definition of quantum t-wise independence;

2. Explicit construction of a t-wise independent set of quantum states.

3. Derandomizing measurements in a random basis.

Part 1

Defining quantum pseudorandomness

Quantum t-designs Sets of quantum states | that are

indistinguishable from Haar measure if we are given access to t copies of |.

Quantum state = unit vector in N complex dimensions.

Haar measure = uniform probability distribution over the unit sphere.

Polynomials A quantum state has the form

Let f()= f(1, 2, …, N) be a degree-t polynomial in the amplitudes.

i

i i

Polynomials Haar measure:

Finite probability distribution

A set of quantum states is a t-design if and only if Ef = Eh, for any polynomial f of degree t.

dfEH

i

iif fpE

Polynomials Haar measure:

Finite probability distribution

If Ef is almost the same as Eh, then the distribution is an approximate t-design.

dfEH

i

if fE

State-of-the art 1-design with N states

(orthonormal basis) 2-designs with O(N2) states (well-

known) t-designs with O(N2t) states

(Kuperberg)

Our contribution

1. Approximate t-designs with O(Nt logc N) states for any t. (Quadratic improvement over previous bound)

2. Derandomization using approximate 4-design.

Part 2

Construction of approximate t-

designs

Step 1 Let f(1, …, N, 1

*, …, N*) be a polynomial of

degree t. We want: a set of states for which E[f] is

almost the same as for random state. Suffices to restrict attention to f a monomial. Further restrict to monomials in 1 and 1

*. Design a probability distribution P1 for 1.

Step 2 For a general monomial f, write

f=f1(i1)…fk(ik

),

If we choose each amplitude i

independently from P1, E[f1] … E[fk] have the right values.

E[f] E[f1] … E[fk].

The problem

If we choose each amplitude independently, there are ~cN possible states

Exponential in the Hilbert space dimension!

t-wise independent distributions Probability distributions over (1,

…, N) in which every set of t coordinates is independent.

Well studied in classical CS. Efficient constructions, with O(Nt)

states.

Step 3 Modify t-wise independent

distribution so that each i is distributed according to P1.

For each (1, …, N), take

Set of O(Nt logcN) quantum states.

i

i i

Final result Theorem Let t>0 be an integer. For

any N, there exists an -approximate t-design in N dimensions with O(NtlogcN) states.

States in the t-design can be efficiently generated.

Application:measurements in a random basis

Task We are given one of two

orthogonal quantum states |0, |1.

Determine if the state is |0 or |1.

Simple solution Measurement

basis that includes |0 and |1.

The other basis vectors are orthogonal to |0 and |1.

|0, |1, |2, …,

|0

0

|1

1

What if we don’t know prior to designing the measurement which states we’ll have to distinguish?

Measurement in a random basis

Let |0, |1 be orthogonal quantum states.

Theorem [Radhakrishnan, et al., 2005] Let M be a random orthonormal basis. Let P0 and P1 be probability distributions

obtained by measuring |0, |1 w.r.t. M. W.h.p., P0 and P1 differ by at least c>0 in

variation distance.

Measurement in a non-random basis

Let |0 and |1 be orthogonal quantum states.

Theorem Let M be an approximate 4-design. Let P0, P1 be the probability distributions

obtained by measuring |0, |1 w.r.t. M.

We always have |P0-P1|>c. Here, |P0-P1|=i|P0(i)-P1(i)|.

Proof sketch

We would like to express |P0-P1| as a polynomial in the amplitudes of the measurement basis.

Problem: |P0-P1| not a polynomial.

Proof sketch Solution is to switch to quantities that are polynomials in the amplitudes: |P0-P1|2

2=i|P0(i)-P1(i)|2 ;

|P0-P1|44=i|P0(i)-P1(i)|4 .

Bounds on |P0-P1|22,|P0-P1|4

4 imply

bound on |P0-P1|. Fourth moment method [Berger, 1989].

Summary Definition of approximate t-designs

for quantum states. Constructions of approximate t-

designs with O(Nt logcN) states. Derandomization for

measurements, using a 4-design (first application of t-designs for t>2 in quantum information).

Open problem t-designs for unitary

transformations? Known constructions for t=1, t=2. Proofs of existence for t>2. No efficient constructions for t>2.