Quantum t-designs: t- wise independence in the quantum world Andris Ambainis, Joseph Emerson IQC, University of Waterloo
Mar 26, 2015
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.