Quantum algorithms vs. polynomials and the maximum quantum-classical gap in the query model

Query model  Function f(x 1,..., x N ), x i  {0,1}.  x i given by a black box: i xixi Complexity = number of queries

Quantum query model  Fixed starting state.  U 0, U 1, …, U T – independent of x 1, …, x N.  Q – queries: U0U0 QQ U1U1 UTUT …

Reasons to study query model  Encompasses many quantum algorithms (Grover’s search, quantum part of factoring, etc.).  Provable quantum-vs-classical gaps.

Quantum vs. classical 1 query quantumly How many queries classically?

Period finding x 1, x 2,..., x N - periodic i xixi Find period r 1 query quantumly

Period-finding  Quantum algorithm works if N  r 2.  T classical queries – can test T 2 possible periods. i xixi queries classically

Our result [Aaronson, A]  Task that requires 1 query quantumly,  (  N) classically.  1 query quantum algorithms can be simulated by O(  N) query probabilistic algorithms.

Fourier checking/Forrelation

Forrelation  Input: (x 1,..., x N, y 1,..., y N )  {-1, 1} 2N.  Are vectors highly correlated? F N – Fourier transform over Z N. F N – Fourier transform over Z N.

More precisely...  Is the inner product  3/5 or  1/100?

Quantum algorithm 1. Generate states in parallel (1 query). 2. Apply F N to 2 nd state. 3. Test if states equal (SWAP test).

Classical lower bound  Theorem Any classical algorithm for FORRELATION uses queries.

REAL FORRELATION  Distinguish between random (x i ’s - Gaussian); random,.  Real-valued vectors

Lower bound  Claim REAL FORRELATION requires queries.  Intuition: if, each variable – Gaussian, correlations between x i ’s and y j ’s - weak.  o(  N) values x i and y j  uncorrelated random variables.

Reduction  Proof idea: Replace x i  sgn(x i ) to achieve x i  {-1, 1}. T query algorithm for FORRELATION T query algorithm for REAL FORRELATION

Simulating 1 query quantum algorithms

Simulation  Theorem Any 1 query quantum algorithm can be simulated probabilistically using O(  N) queries.

Analyzing query algorithms QQ QUTUT … U1U1  1,1 |1,1  +  1,2 |1, 2  + … +  N, M |N, M   1,1 is actually  1,1 (x 1,..., x N )

Polynomials method  Lemma [Beals et al., 1998] After k queries, the amplitudes are polynomials in x 1,..., x N of degree  k. Measurement: Polynomial of degree  2k

Our task  Pr[A outputs 1] = p(x 1,..., x N ), deg p =2.  0  p(x 1,..., x N )  1.  Task: estimate p(x 1,..., x N ) with precision . Solution: random sampling.

Pre-processing  Problem: large error if sampling omits x i with large influence in p(x 1,..., x N ).  Solution: replace influential x i ’s by several variables with smaller influence.

Sampling 1 ample N of N 2 terms independently. Good if we sample N of N 2 terms independently. Estimator: Requires sampling N variables x i !

Sampling 2 Sampling N terms a i,j x i x j Sampling  N variables x i 

Extension to k queries  Theorem k query quantum algorithms can be simulated probabilistically with O(N 1-1/2k ) queries.  Proof:  Algorithm  polynomial of degree 2k;  Random sampling.  Question: Is this optimal?

K-fold forrelation

 Forrelation: given black box functions f(x) and g(y), estimate  K-fold forrelation: given f 1 (x),..., f k (x), estimate

Results  Theorem k-fold forrelation can be solved with  k/2  quantum queries.  Conjecture k-fold forrelation requires  (N 1-1/k ) queries classically.

From polynomials to quantum algorithms (with Scott Aaronson, Jānis Iraids, Mārtiņš Kokainis, Juris Smotrovs)

Quantum algorithm with t queries Polynomials of degree 2t ??

Quantum algorithm with 1 query Polynomials of degree 2  Our result

More precisely...  Polynomial p represents f with error  if: f = 0  p  [0,  ]; f = 1  p  [1- , 1]; f – undefined  p  [0, 1].  Theorem Q  (f)=1 for some  <1/2 iff f can be represented by p: deg p=2 with error  <1/2.

Standard polynomial representation Block-multilinear representation Step 1

Requirements  q(x 1,..., x N, x 1,..., x N )  f(x 1,..., x N );  q(x 1,..., x N, y 1,..., y N )  [-1, 1] for all x i, y j  {0, 1}.

Step 2: evaluating q  U = (N  a i,j ) – unitary.  SWAP test on |  x  and U|  y  : Still works if ||U||  C!

Two norms Have: |q|  1 Need:

Step 3: variable splitting  Replace x i by, - new variables. 1. |q|  1 preserved; 2. Influential variables - eliminated.

Result Variable-splitting K – Groethendieck’s constant

Summary  1 quantum query =  (  N) classical queries.  k quantum queries can be simulated with O(N 1-1/2k ) classical queries.  1 quantum query = polynomials of degree 2.

Open problem 1  Does k-fold FORRELATION require  (N 1-1/2k ) queries classically?  Plausible but looks quite difficult matematically.

Open problem 2  Best quantum-classical gaps: 1 quantum query -  (  N) classical queries; 2 quantum queries -  (  N) classical queries;... log N quantum queries - classical queries. Any problem that requires O(log N) queries quantumly,  (N c ), c>1/2 classically?

Open problem 3  Characterize quantum algorithms with 2, 3,..., queries?  2 queries  polynomials of degree 4?  Polynomials of degree 3  2 query algorithms?