Quantum Counterfeit Coin Problems Kazuo Iwama (Kyoto Univ.) Harumichi Nishimura (Osaka Pref. Univ.) Rudy Raymond (IBM Research - Tokyo) Junichi Teruyama (Kyoto Univ.) ISAAC2010, Dec.15, 2010, Jeju (Korea)

Speedup by Quantum Algorithms Superpolynomial (or more)Quadratic (or less) Early-day's algorithm  Bernstein-Vazirani  Simon Shor's algorithm  Integer factoring  Discrete logarithm Shor's extentions  Pell's equation  Hidden subgroup problems Quantum walk algorithm  Glued trees graph Grover's algorithm  Unordered search Grover's application  Amplitude amplification  Quantum counting Quantum walk algorithm  Element distinctness  Graph problems (e.g., Triangle finding)  Spatial search (e.g., 2D-grid)  NAND tree Our algorithm achieves quartic speedup

Counterfeit Coin Problem You have eight similar coins and a beam balance. At most one coin is counterfeit and hence underweight. How can you detect whether there is an underweight coin, and if so, which one, using the balance only twice? The counterfeit coin problem is a well-known puzzle. [E.Schell, American Mathematical Monthly 52, p.46, 1945]

Counterfeit Coin Problem leans to Right Balanced leans to Left R 12 L B 3 R 45 L B 6 R 78 L ←Answer

Counterfeit Coin Problem There have been several different versions and extensions in the literature, say, more counterfeit coins, whether the counterfeit coins are underweight or overweight, they have the equal weight or not, and so on. Here, we assume: k false coins with equal weight are included in N given coins. The balance scale gives us only binary information, balanced (i.e., two sets of coins on the pans are equal in weight) or titled (different in weight) The goal is to find all the k false coins.

B-Oracle (Balance Oracle) Model Our setting is naturally considered as an oracle model : (Unknown) input : N bits Query string : N trits including the same number of 1's and (-1)'s Answer : 1 bit where (balanced) (tilted) B-oracle Ex

B-Oracle (Balance Oracle) Model Our setting is naturally considered as an oracle model : (Unknown) input : N bits Query string : N trits including the same number of 1's and (-1)'s Answer : 1 bit where (balanced) (tilted) When k=1, the query complexity (of finding the false coin) is log 2 N, which is also an information theoretic lower bound. B-oracle

Quantum Counterfeit Coin Problem Q. How about the quantum version of the counterfeit coin problem? The B-oracle model can be naturally quantized: We can identify what the oracle is if all final states are orthogonal. orthogonal = distinguishable

Quantum Counterfeit Coin Problem Q. What is the quantum query complexity of finding all the k false coins, that is, identifying the input x? Our goal is to answer the following question: k=1k=2k=3 general Quantum 112≤ k ≤3O(k 1/4 ) Classical log N ≥2log(N/2)≥3log(N/3) Ω(k log(N/k)) quartic speed-up (Note) So far, there are few natural problems whose quantum speed-up are between quadratic and exponential (ex. cubic [van Dam-Shparlinski 08]). Results

Algorithm

B-Oracle and IP oracle Notice a similarity between the B-oracle and the IP (Inner Product) oracle! (Unknown) input : N bits Query string : N trits including the same number of 1's and (-1)'s Answer : 1 bit where (balanced) (tilted) B-oracle (Unknown) input : N bits Query string : N bits Answer : 1 bit IP oracle

Case: k=1 So, we can identify the oracle by only 1 query! query to B-oracle under a reversible operation When k=1 (and the Hamming weight of the query string is even), the B-oracle can simulate IP oracle! The first half of the nonzero entries are -1 and the last half are 1 Key Fact orthogonal = distinguishable Bernstein-Vazirani 1997 [cf. Terhal-Smolin 1998]

General k In general, B-oracle and IP oracle are much different. But, we can still simulate IP oracle by O(k 1/4 ) queries to B-oracle! Find balanced pairs by the Grover search (exactly, amplitude amplification) If the Grover search finds a solution (=balanced pair), do nothing. Otherwise flip the phase. basis change Algorithm: Find*(k)

Lower bounds

Lower Bounds under Restricted Pan-size L :=the size of the pans (=the number of coins placed on each pan) L ≥ l ( L ≤ l, resp.) denotes the restriction that we should place at least l (at most l, resp.) coins the pans whenever we use the balance. [Lower bound under Restricted Pan-size] If L ≤ N/k 1+2ε or L ≥ N/k 1-4ε, then Ω(k ε ) queries are needed. In particular, if L ≤ O(N/k 1.5 ) or L ≥ Ω(N), then tight lower bound Ω(k 1/4 ) can be obtained. (Note) Algorithm Find*(k) is easily modified so that L ≥ Ω(N) can be satisfied. or

For Quantum Lower Bounds Two standard techniques Polynomial methods [Beals-Buhrman-Cleve-Mosca-de Wolf 1998] Adversary methods [Ambainis 2000] orthogonal = distinguishable =1

Lower Bounds under Restricted Pan-size L :=the size of the pans (=the number of coins placed on each pan) L ≥ l ( L ≤ l, resp.) denotes the restriction that we should place at least l (at most l, resp.) coins the pans whenever we use the balance. [Lower bound under Restricted Pan-size] If L ≤ N/k 1+2ε or L ≥ N/k 1-4ε, then Ω(k ε ) queries are needed. In particular, if L ≤ O(N/k 1.5 ) or L ≥ Ω(N), then tight lower bound Ω(k 1/4 ) can be obtained. (Note) Algorithm Find*(k) is easily modified so that L ≥ Ω(N) can be satisfied. placing few coins gives little information placing many coins gives little information l N/k 1/5 k 1/5

More Witness on the tightness of the O(k 1/4 )-Algorithm (Informal Statement) If the use of B-oracle is restricted so that the set of coins placed on the two pans can be partitioned into the left pan and right pan uniformly at random, we need Ω(k 1/4 ) queries. [Lower bound under "random-partition assumption" ] To show this statement rigorously, need to extend an oracle operation in a ''stochastic'' form extend an adversary method to the "stochastic'' oracle Notice that our algorithm is always used in such a way that the partition of the coins into the two pans is done uniformly at random. There seems no essentially better ways than this when using the B-oracle...

Conclusion We have investigated the quantum query complexity of finding k false coins from N coins by using the B-oracle, which represents a balance scale. We have obtained upper bound O(k 1/4 ), contrasting with the classical lower bound Ω(klog(N/k)). So this achieves quartic speed-up for a natural problem. We do not have a matching lower bound, but we have obtained several tight lower bounds under different restrictions of the ways that algorithms can take. At least, they imply that we need a radically new algorithm to beat the current upper bound.