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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.
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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 Eve | • 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 |. B A
Private quantum channels | Ui| | [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|. B A
Private quantum channels | Ui| | [Hayden et al., 2001]: • If U1, U2, …are uniformly random unitary transformations, Eve gets almost no information about |. B A
Summary • Random quantum objects are useful! • How do we generate and describe a random state? • A random state on n qubits has 2n amplitudes. • Sinceamplitudes 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 • Definition of quantum t-wise independence; • Explicit construction of a t-wise independent set of quantum states. • 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.
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.
Polynomials • Haar measure: • Finite probability distribution • If Ef is almost the same as Eh, then the distribution is an approximate t-design.
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 • Approximate t-designs with O(Nt logc N) states for any t. (Quadratic improvement over previous bound) • 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 • If we choose each amplitude i independently from P1, E[f1] … E[fk] have the right values. • For a general monomial f, write f=f1(i1)…fk(ik), 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.
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.
Task • We are given one of two orthogonal quantum states |0, |1. • Determine if the state is |0 or |1.
|0 |1 0 1 Simple solution • Measurement basis that includes |0 and |1. • The other basis vectors are orthogonal to |0 and |1. |0, |1, |2, …, 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|22=i|P0(i)-P1(i)|2 ; |P0-P1|44=i|P0(i)-P1(i)|4 . • Bounds on |P0-P1|22,|P0-P1|44imply 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.
