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arXiv:quant-ph/0304131

Quantum Evaluation of Multi-Valued Boolean Functions

Kazuo Iwama, Akinori Kawachi, Hiroyuki Masuda, Putra, Raymond H., Shigeru Yamashita

Published Apr 21, 2003• 1 citation
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Abstract

Our problem is to evaluate a multi-valued Boolean function $F$ through oracle calls. If $F$ is one-to-one and the size of its domain and range is the same, then our problem can be formulated as follows: Given an oracle $f(a,x): \{0,1\}^n\times\{0,1\}^n \to \{0,1\}$ and a fixed (but hidden) value $a_0$, we wish to obtain the value of $a_0$ by querying the oracle $f(a_0,x)$. Our goal is to minimize the number of such oracle calls (the query complexity) using a quantum mechanism. Two popular oracles are the EQ-oracle defined as $f(a,x)=1$ iff $x=a$ and the IP-oracle defined as $f(a,x)= a\cdot x \mod 2$. It is also well-known that the query complexity is $Θ(\sqrt{N})$ ($N=2^n$) for the EQ-oracle while only O(1) for the IP-oracle. The main purpose of this paper is to fill this gap or to investigate what causes this large difference. To do so, we introduce a parameter $K$ as the maximum number of 1's in a single column of $T_f$ where $T_f$ is the $N\times N$ truth-table of the oracle $f(a,x)$. Our main result shows that the (quantum) query complexity is heavily governed by this parameter $K$: ($i$) The query complexity is $Ω(\sqrt{N/K})$. ($ii$) This lower bound is tight in the sense that we can construct an explicit oracle whose query complexity is $O(\sqrt{N/K})$. ($iii$) The tight complexity, $Θ(\frac{N}{K}+\log{K})$, is also obtained for the classical case. Thus, the quantum algorithm needs a quadratically less number of oracle calls when $K$ is small and this merit becomes larger when $K$ is large, e.g., $\log{K}$ v.s. constant when $K = cN$.

DOI
10.48550/arxiv.quant-ph/0304131

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Our problem is to evaluate a multi-valued Boolean function FFF through oracle calls. If FFF is one-to-one and the size of its domain and range is the same, then our problem can be formulated as follows: Given an oracle f(a,x):{0,1}n×{0,1}n→{0,1}f(a,x): \{0,1\}^n\times\{0,1\}^n \to \{0,1\}f(a,x):{0,1}n×{0,1}n→{0,1} and a fixed (but hidden) value a0a_0a0​, we wish to obtain the value of a0a_0a0​ by querying the oracle f(a0,x)f(a_0,x)f(a0​,x). Our goal is to minimize the number of such oracle calls (the query complexity) using a quantum mechanism. Two popular oracles are the EQ-oracle defined as f(a,x)=1f(a,x)=1f(a,x)=1 iff x=ax=ax=a and the IP-oracle defined as f(a,x)=a⋅xmod  2f(a,x)= a\cdot x \mod 2f(a,x)=a⋅xmod2. It is also well-known that the query complexity is Θ(N)Θ(\sqrt{N})Θ(N​) (N=2nN=2^nN=2n) for the EQ-oracle while only O(1) for the IP-oracle. The main purpose of this paper is to fill this gap or to investigate what causes this large difference. To do so, we introduce a parameter KKK as the maximum number of 1's in a single column of TfT_fTf​ where TfT_fTf​ is the N×NN\times NN×N truth-table of the oracle f(a,x)f(a,x)f(a,x). Our main result shows that the (quantum) query complexity is heavily governed by this parameter KKK: (iii) The query complexity is Ω(N/K)Ω(\sqrt{N/K})Ω(N/K​). (iiiiii) This lower bound is tight in the sense that we can construct an explicit oracle whose query complexity is O(N/K)O(\sqrt{N/K})O(N/K​). (iiiiiiiii) The tight complexity, Θ(NK+log⁡K)Θ(\frac{N}{K}+\log{K})Θ(KN​+logK), is also obtained for the classical case. Thus, the quantum algorithm needs a quadratically less number of oracle calls when KKK is small and this merit becomes larger when KKK is large, e.g., log⁡K\log{K}logK v.s. constant when K=cNK = cNK=cN.
In this paper, we introduce the hybrid query complexity, denoted as Q(f;q)\mathrm{Q}(f;q)Q(f;q), which is the minimal query number needed to compute fff, when a classical decision tree is allowed to call q′q'q′-query quantum subroutines for any q′≤qq'\leq qq′≤q. We present the following results: ∙\bullet∙ There exists a total Boolean function fff such that Q(f;1)=O~(R(f)4/5)\mathrm{Q}(f;1) = \widetilde{\mathcal{O}}(\mathrm{R}(f)^{4/5})Q(f;1)=O(R(f)4/5). ∙\bullet∙ Q(f;q)=Ω(bs(f)/q+bs(f))\mathrm{Q}(f;q) = Ω(\mathrm{bs}(f)/q + \sqrt{\mathrm{bs}(f)})Q(f;q)=Ω(bs(f)/q+bs(f)​) for any Boolean function fff; the lower bound is tight when fff is the OR{\rm O{\small R}}OR function. ∙\bullet∙ Q(g∘XORClog⁡n;1)=Ω~(n)\mathrm{Q}(g \circ {\rm X{\small OR}}_{C \log n};1) = \widetildeΩ(\sqrt{n})Q(g∘XORClogn​;1)=Ω(n​) for some sufficiently large constant CCC, where g:=BOOLSIMONng := {\rm B{\small OOL}S{\small IMON}}_ng:=BOOLSIMONn​ is a variant of Simon's problem. Note that Q(g∘XORClog⁡n)=O(polylog  n)\mathrm{Q}(g\circ {\rm X{\small OR}}_{C \log n}) = \mathcal{O}(\mathrm{polylog}\; n)Q(g∘XORClogn​)=O(polylogn). Therefore an exponential separation is established. Furthermore, this open the road to prove the conjecture ∀k, Q(g∘XORClog⁡k+1n;log⁡kn)=Ω~(n)\forall k,\,\mathrm{Q}(g \circ {\rm X{\small OR}}_{C \log^{k+1} n};\log^{k} n) = \widetildeΩ(\sqrt{n})∀k,Q(g∘XORClogk+1n​;logkn)=Ω(n​), which would imply the oracle separation HP(QSIZE(nα))O⊊BQPO\mathsf{HP}(\mathsf{QSIZE}(n^α))^\mathfrak{O} \subsetneq \mathsf{BQP}^\mathfrak{O}HP(QSIZE(nα))O⊊BQPO for any ααα, where HP(QSIZE(nα))\mathsf{HP}(\mathsf{QSIZE}(n^α))HP(QSIZE(nα)) is a complexity class that contains BQTIME(nα)BPP\mathsf{BQTIME}(n^α)^{\mathsf{BPP}}BQTIME(nα)BPP and BPPBQTIME(nα)\mathsf{BPP}^{\mathsf{BQTIME}(n^α)}BPPBQTIME(nα) in any relativized world.
Given a Boolean function f:{0,1}n→{0,1}f:\{0,1\}^n\to\{0,1\}f:{0,1}n→{0,1}, the goal in the usual query model is to compute fff on an unknown input x∈{0,1}nx \in \{0,1\}^nx∈{0,1}n while minimizing the number of queries to xxx. One can also consider a "distinguishing" problem denoted by fsabf_{\mathsf{sab}}fsab​: given an input x∈f−1(0)x \in f^{-1}(0)x∈f−1(0) and an input y∈f−1(1)y \in f^{-1}(1)y∈f−1(1), either all differing locations are replaced by a ∗*∗, or all differing locations are replaced by †\dagger†, and an algorithm's goal is to identify which of these is the case while minimizing the number of queries. Ben-David and Kothari [ToC'18] introduced the notion of randomized sabotage complexity of a Boolean function to be the zero-error randomized query complexity of fsabf_{\mathsf{sab}}fsab​. A natural follow-up question is to understand Q(fsab)\mathsf{Q}(f_{\mathsf{sab}})Q(fsab​), the quantum query complexity of fsabf_{\mathsf{sab}}fsab​. In this paper, we initiate a systematic study of this. The following are our main results: ∙    \bullet\;\;∙ If we have additional query access to xxx and yyy, then Q(fsab)=O(min⁡{Q(f),n})\mathsf{Q}(f_{\mathsf{sab}})=O(\min\{\mathsf{Q}(f),\sqrt{n}\})Q(fsab​)=O(min{Q(f),n​}). ∙    \bullet\;\;∙ If an algorithm is also required to output a differing index of a 0-input and a 1-input, then Q(fsab)=O(min⁡{Q(f)1.5,n})\mathsf{Q}(f_{\mathsf{sab}})=O(\min\{\mathsf{Q}(f)^{1.5},\sqrt{n}\})Q(fsab​)=O(min{Q(f)1.5,n​}). ∙    \bullet\;\;∙ Q(fsab)=Ω(fbs(f))\mathsf{Q}(f_{\mathsf{sab}}) = Ω(\sqrt{\mathsf{fbs}(f)})Q(fsab​)=Ω(fbs(f)​), where fbs(f)\mathsf{fbs}(f)fbs(f) denotes the fractional block sensitivity of fff. By known results, along with the results in the previous bullets, this implies that Q(fsab)\mathsf{Q}(f_{\mathsf{sab}})Q(fsab​) is polynomially related to Q(f)\mathsf{Q}(f)Q(f). ∙    \bullet\;\;∙ The bound above is easily seen to be tight for standard functions such as And, Or, Majority and Parity. We show that when fff is the Indexing function, Q(fsab)=Θ(fbs(f))\mathsf{Q}(f_{\mathsf{sab}})=Θ(\mathsf{fbs}(f))Q(fsab​)=Θ(fbs(f)), ruling out the possibility that Q(fsab)=Θ(fbs(f))\mathsf{Q}(f_{\mathsf{sab}})=Θ(\sqrt{\mathsf{fbs}(f)})Q(fsab​)=Θ(fbs(f)​) for all fff.
Studies the quantum complexity of the static set membership problem: given a subset S (|S|/spl les/n) of a universe of size m(/spl Gt/n), store it as a table, T:(0,1)/sup r//spl rarr/(0,1), of bits so that queries of the form 'is x in S?' can be answered. The goal is to use a small table and yet answer queries using a few bit probes. This problem was considered by H. Buhrman et al. (2000), who showed lower and upper bounds for this problem in the classical deterministic and randomised models. In this paper, we formulate this problem in the "quantum bit-probe model". We assume that access to the table T is provided by means of a black-box (oracle) unitary transform O/sub T/ that takes the basis state (y,b) to the basis state |y,b/spl oplus/T(y)>. The query algorithm is allowed to apply O/sub T/ on any superposition of basis states. We show tradeoff results between the space (defined as 2/sup r/) and the number of probes (oracle calls) in this model. Our results show that the lower bounds shown by Buhrman et al. for the classical model also hold (with minor differences) in the quantum bit-probe model. These bounds almost match the classical upper bounds. Our lower bounds are proved using linear algebraic arguments.
The oracle identification problem (OIP) was introduced by Ambainis et al. \cite{AIKMRY04}. It is given as a set SSS of MMM oracles and a blackbox oracle fff. Our task is to figure out which oracle in SSS is equal to the blackbox fff by making queries to fff. OIP includes several problems such as the Grover Search as special cases. In this paper, we improve the algorithms in \cite{AIKMRY04} by providing a mostly optimal upper bound of query complexity for this problem: (iii) For any oracle set SSS such that ∣S∣≤2Nd|S| \le 2^{N^d}∣S∣≤2Nd (d<1d < 1d<1), we design an algorithm whose query complexity is O(Nlog⁡M/log⁡N)O(\sqrt{N\log{M}/\log{N}})O(NlogM/logN​), matching the lower bound proved in \cite{AIKMRY04}. (iiiiii) Our algorithm also works for the range between 2Nd2^{N^d}2Nd and 2N/log⁡N2^{N/\log{N}}2N/logN (where the bound becomes O(N)), but the gap between the upper and lower bounds worsens gradually. (iiiiiiiii) Our algorithm is robust, namely, it exhibits the same performance (up to a constant factor) against the noisy oracles as also shown in the literatures \cite{AC02,BNRW03,HMW03} for special cases of OIP.
Quantum oracle of Boolean functions is one of the central bridges between classical and quantum algorithms, but the study focusing at quantum circuit optimization of such oracle is yet closed. In this paper, we propose nearly optimal tradeoffs among circuit size, circuit depth and ancilla count, for quantum oracles of three kinds of Boolean functions: general total Boolean functions with output size bbb: with 1≤m≤Θ(2nn)1\le m\leΘ\left(\frac{2^n}{n}\right)1≤m≤Θ(n2n​) ancilla, size O(b2nlog⁡(n+m))\mathcal{O}\left(\frac{b2^n}{\log(n+m)}\right)O(log(n+m)b2n​), depth O(b2nn+m)\mathcal{O}\left(\frac{b2^n}{n+m}\right)O(n+mb2n​); partial Boolean functions of effective support size ddd and output size bbb: with Θ(log⁡d)≤m≤Θ(d)Θ\left(\log d\right)\le m\le Θ\left(d\right)Θ(logd)≤m≤Θ(d) ancilla, size O(nlog⁡d+bd)\mathcal{O}\left(n\log d+bd\right)O(nlogd+bd), depth O(nlog⁡nlog⁡dn+m+log⁡n+d(log⁡d+blog⁡m)m)\mathcal{O}\left(\frac{n\log n\log d}{n+m}+\log n+\frac{d(\log d+b\log m)}{m}\right)O(n+mnlognlogd​+logn+md(logd+blogm)​); sparse Boolean functions of true input size ddd: with Θ(log⁡n+log⁡d)≤m≤Θ(ndlog⁡d)Θ\left(\log n+\log d\right)\le m\leΘ\left(\frac{nd}{\log d}\right)Θ(logn+logd)≤m≤Θ(logdnd​) ancilla, size O(n2log⁡d+ndlog⁡(log⁡d+m/n))\mathcal{O}\left(n^2\log d+\frac{nd}{\log(\log d+m/n)}\right)O(n2logd+log(logd+m/n)nd​), depth O(n2log⁡nlog⁡dn+m+log⁡n+ndm)\mathcal{O}\left(\frac{n^2\log n\log d}{n+m}+\log n+\frac{nd}{m}\right)O(n+mn2lognlogd​+logn+mnd​). All the size and depth bounds are asymptotically optimal up to logarithmic factors in the corresponding ancilla count regions. We hope these results find applications in scenarios where classical procedures are needed to be embedded into quantum circuits, such as QROM implementation and quantum algorithm design.
The degree of a polynomial representing (or approximating) a function f is a lower bound for the quantum query complexity of f. This observation has been a source of many lower bounds on quantum algorithms. It has been an open problem whether this lower bound is tight. This is why Boolean functions are needed with a high number of essential variables and a low polynomial degree. Unfortunately, it is a well-known problem to construct such functions. The best separation between these two complexity measures of a Boolean function was exhibited by Ambai- nis [5]. He constructed functions with polynomial degree M and number of variables Omega(M2). We improve such a separation to become exponential. On the other hand, we use a computerized exhaustive search to prove tightness of this bound.