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

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Random quantum states Several recent results using random quantum objects: Random quantum states; Random unitary transformations; Random orthonormal bases.

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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

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Private quantum channels [Hayden et al., 2001]: Let N = dim |. Let U 1, U 2, … be O(N log N) unitaries, known to both Alice and Bob. Alice randomly chooses U i, sends U i |. A B | U i | |

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Private quantum channels [Hayden et al., 2001]: If U 1, U 2, …are uniformly random unitary transformations, Eve gets almost no information about |. A B | U i | |

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Summary Random quantum objects are useful! How do we generate and describe a random state? A random state on n qubits has 2 n amplitudes. Since amplitudes are random, 2 n are bits required to describe the state. Protocols are highly inefficient!

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Quantum pseudorandomness We want small sets of quantum states, with properties similar to random states. In this talk: quantum counterpart of t- wise independence.

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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.

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Part 1 Defining quantum pseudorandomness

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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.

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Polynomials A quantum state has the form Let f( )= f( 1, 2, …, N ) be a degree-t polynomial in the amplitudes.

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Polynomials Haar measure: Finite probability distribution A set of quantum states is a t-design if and only if E f = E h, for any polynomial f of degree t.

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Polynomials Haar measure: Finite probability distribution If E f is almost the same as E h, then the distribution is an approximate t-design.

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State-of-the art 1-design with N states (orthonormal basis) 2-designs with O(N 2 ) states (well- known) t-designs with O(N 2t ) states (Kuperberg)

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Our contribution 1.Approximate t-designs with O(N t log c N) states for any t. (Quadratic improvement over previous bound) 2.Derandomization using approximate 4- design.

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Part 2 Construction of approximate t-designs

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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 P 1 for 1.

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Step 2 For a general monomial f, write f=f 1 ( i 1 )…f k ( i k ), If we choose each amplitude i independently from P 1, E[f 1 ] … E[f k ] have the right values. E[f] E[f 1 ] … E[f k ].

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The problem If we choose each amplitude independently, there are ~c N possible states Exponential in the Hilbert space dimension!

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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(N t ) states.

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Step 3 Modify t-wise independent distribution so that each i is distributed according to P 1. For each ( 1, …, N ), take Set of O(N t log c N) quantum states.

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Final result Theorem Let t>0 be an integer. For any N, there exists an -approximate t- design in N dimensions with O(N t log c N) states. States in the t-design can be efficiently generated.

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Application:measurements in a random basis

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Task We are given one of two orthogonal quantum states | 0, | 1. Determine if the state is | 0 or | 1.

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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 dont know prior to designing the measurement which states well have to distinguish?

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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 P 0 and P 1 be probability distributions obtained by measuring | 0, | 1 w.r.t. M. W.h.p., P 0 and P 1 differ by at least c>0 in variation distance.

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Measurement in a non-random basis Let | 0 and | 1 be orthogonal quantum states. Theorem Let M be an approximate 4-design. Let P 0, P 1 be the probability distributions obtained by measuring | 0, | 1 w.r.t. M. We always have |P 0 -P 1 |>c. Here, |P 0 -P 1 |= i |P 0 (i)-P 1 (i)|.

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Proof sketch We would like to express |P 0 -P 1 | as a polynomial in the amplitudes of the measurement basis. Problem: |P 0 -P 1 | not a polynomial.

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Proof sketch Solution is to switch to quantities that are polynomials in the amplitudes: |P 0 -P 1 | 2 2 = i |P 0 (i)-P 1 (i)| 2 ; |P 0 -P 1 | 4 4 = i |P 0 (i)-P 1 (i)| 4. Bounds on |P 0 -P 1 | 2 2,|P 0 -P 1 | 4 4 imply bound on |P 0 -P 1 |. Fourth moment method [Berger, 1989].

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Summary Definition of approximate t-designs for quantum states. Constructions of approximate t-designs with O(N t log c N) states. Derandomization for measurements, using a 4-design (first application of t- designs for t>2 in quantum information).

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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.

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