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arXiv:1606.04016

Adaptivity vs Postselection

Lijie Chen

Published Jun 13, 2016• 2 citations
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Abstract

We study the following problem: with the power of postselection (classically or quantumly), what is your ability to answer adaptive queries to certain languages? More specifically, for what kind of computational classes $\mathcal{C}$, we have $\mathsf{P}^{\mathcal{C}}$ belongs to $\mathsf{PostBPP}$ and $\mathsf{PostBQP}$? While a complete answer to the above question seems impossible given the development of present computational complexity theory. We study the analogous question in query complexity, which sheds light on the limitation of {\em relativized} methods (the relativization barrier) to the above question. Informally, we show that, for a partial function $f$, if there is no efficient (In the world of query complexity, being efficient means using $O(\operatorname*{polylog}(n))$ time.) {\em small bounded-error} algorithm for $f$ classically or quantumly, then there is no efficient postselection bounded-error algorithm to answer adaptive queries to $f$ classically or quantumly. Our results imply a new proof for the classical oracle separation $\mathsf{P}^{\mathsf{NP}^{\mathcal{O}}} \not\subset \mathsf{PP}^{\mathcal{O}}$. They also lead to a new oracle separation $\mathsf{P}^{\mathsf{SZK}^{\mathcal{O}}} \not\subset \mathsf{PP}^{\mathcal{O}}$. Our result also implies a hardness amplification construction for polynomial approximation: given a function $f$ on $n$ bits, we construct an adaptive-version of $f$, denoted by $F$, on $O(m \cdot n)$ bits, such that if $f$ requires large degree to approximate to error $2/3$ in a certain one-sided sense, then $F$ requires large degree to approximate even to error $1/2 - 2^{-m}$. Our construction achieves the same amplification in the work of Thaler (ICALP, 2016), by composing a function with $O(\log n)$ {\em deterministic query complexity}.

DOI
10.48550/arxiv.1606.04016

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Post-selected Classical Query Complexity

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We study classical query algorithms with post-selection, and find that they are closely connected to rational functions with nonnegative coefficients. We show that the post-selected classical query complexity of a Boolean function is equal to the minimal degree of a rational function with nonnegative coefficients that approximates it (up to a factor of two). For post-selected quantum query algorithms, a similar relationship was shown by Mahadev and de Wolf, where the rational approximations are allowed to have negative coefficients. Using our characterisation, we find an exponentially large separation between post-selected classical query complexity and post-selected quantum query complexity, by proving a lower bound on the degree of rational approximations (with nonnegative coefficients) to the Majority function. This lower bound can be generalised to arbitrary symmetric functions, and allows us to find an unbounded separation between non-deterministic quantum and post-selected classical query complexity. All lower bounds carry over into the communication complexity setting. We show that the zero-error variants of post-selected query algorithms are equivalent to non-deterministic classical query algorithms, which in turn are characterised by nonnegative polynomials, and for some problems require exponentially more queries to the input than their bounded-error counterparts. Finally, we describe a post-selected query algorithm for approximating the Majority function, and an efficient query algorithm for approximate counting.

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We study the following problem: with the power of postselection (classically or quantumly), what is your ability to answer adaptive queries to certain languages? More specifically, for what kind of computational classes C\mathcal{C}C, we have PC\mathsf{P}^{\mathcal{C}}PC belongs to PostBPP\mathsf{PostBPP}PostBPP and PostBQP\mathsf{PostBQP}PostBQP? While a complete answer to the above question seems impossible given the development of present computational complexity theory. We study the analogous question in query complexity, which sheds light on the limitation of {\em relativized} methods (the relativization barrier) to the above question. Informally, we show that, for a partial function fff, if there is no efficient (In the world of query complexity, being efficient means using O(polylog⁡(n))O(\operatorname*{polylog}(n))O(polylog(n)) time.) {\em small bounded-error} algorithm for fff classically or quantumly, then there is no efficient postselection bounded-error algorithm to answer adaptive queries to fff classically or quantumly. Our results imply a new proof for the classical oracle separation PNPO⊄PPO\mathsf{P}^{\mathsf{NP}^{\mathcal{O}}} \not\subset \mathsf{PP}^{\mathcal{O}}PNPO⊂PPO. They also lead to a new oracle separation PSZKO⊄PPO\mathsf{P}^{\mathsf{SZK}^{\mathcal{O}}} \not\subset \mathsf{PP}^{\mathcal{O}}PSZKO⊂PPO. Our result also implies a hardness amplification construction for polynomial approximation: given a function fff on nnn bits, we construct an adaptive-version of fff, denoted by FFF, on O(m⋅n)O(m \cdot n)O(m⋅n) bits, such that if fff requires large degree to approximate to error 2/32/32/3 in a certain one-sided sense, then FFF requires large degree to approximate even to error 1/2−2−m1/2 - 2^{-m}1/2−2−m. Our construction achieves the same amplification in the work of Thaler (ICALP, 2016), by composing a function with O(log⁡n)O(\log n)O(logn) {\em deterministic query complexity}.
We study the following problem: with the power of postselection (classically or quantumly), what is your ability to answer adaptive queries to certain languages? More specifically, for what kind of computational classes C, we have P^C belongs to PostBPP or PostBQP? While a complete answer to the above question seems impossible given the development of present computational complexity theory. We study the analogous question in query complexity, which sheds light on the limitation of relativized methods (the relativization barrier) to the above question. Informally, we show that, for a partial function f, if there is no efficient small bounded-error algorithm for f classically or quantumly, then there is no efficient postselection bounded-error algorithm to answer adaptive queries to f classically or quantumly. Our results imply a new proof for the classical oracle separation P^{NP^O} notsubset PP^O, which is arguably more elegant. They also lead to a new oracle separation P^{SZK^O} notsubset PP^O, which is close to an oracle separation between SZK and PP - an open problem in the field of oracle separations. Our result also implies a hardness amplification construction for polynomial approximation: given a function f on n bits, we construct an adaptive-version of f, denoted by F, on O(m·n) bits, such that if f requires large degree to approximate to error 2/3 in a certain one-sided sense, then F requires large degree to approximate even to error 1/2 - 2^{-m}. Our construction achieves the same amplification in the work of Thaler (ICALP, 2016), by composing a function with O(log n) deterministic query complexity, which is in sharp contrast to all the previous results where the composing amplifiers are all hard functions in a certain sense.
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.
In 2009, using the Fourier Checking\textsf{Fourier Checking}Fourier Checking problem, Aaronson claimed to construct the relativized worlds such that BQP⊄BPPpath\textsf{BQP} \not\subset \mathsf{BPP_{path}}BQP⊂BPPpath​ and BQP⊄SZK\textsf{BQP} \not\subset \textsf{SZK}BQP⊂SZK. However, there are subtle errors in the original proof. In this paper, we point out the issues, and rescue these two separations by using more sophisticated constructions. Meanwhile, we take the opportunity to study the complexity classes BPPpath\mathsf{BPP_{path}}BPPpath​ and SZK\textsf{SZK}SZK. We give general ways to construct functions which are hard for SZK\textsf{SZK}SZK and BPPpath\mathsf{BPP_{path}}BPPpath​ (in the query complexity sense). Using these techniques, we give alternative construction for the oracle separation BQP⊄SZK\textsf{BQP} \not\subset \textsf{SZK}BQP⊂SZK, using only Simon's problem. We also give new oracle separations for PSZK\textsf{P}^{\textsf{SZK}}PSZK from BPPpath\mathsf{BPP_{path}}BPPpath​ and PSZK\textsf{P}^{\textsf{SZK}}PSZK from QSK\textsf{QSK}QSK. The latter result suggests that PSZK\textsf{P}^{\textsf{SZK}}PSZK might be strictly larger than SZK\textsf{SZK}SZK.
In 2004, Aaronson introduced the complexity class PostBQP\mathsf{PostBQP}PostBQP (BQP\mathsf{BQP}BQP with postselection) and showed that it is equal to PP\mathsf{PP}PP. Following their line of work, we introduce two new complexity classes. The first, CorrBQP\mathsf{CorrBQP}CorrBQP, is a modification of BQP\mathsf{BQP}BQP which has the power to perform correlated measurements, i.e. measurements that output the same value across a partition of registers. The second, MajBQP\mathsf{MajBQP}MajBQP, augments BQP\mathsf{BQP}BQP with the ability to collapse a register to its most likely measurement outcome. Specifically, we consider two variants, MajBQP\mathsf{MajBQP}MajBQP and AdMajBQP\mathsf{AdMajBQP}AdMajBQP, where the latter may perform intermediate measurements. We exactly characterize the computational power of the models, CorrBQP=AdMajBQP=BPPPP\mathsf{CorrBQP} = \mathsf{AdMajBQP} = \mathsf{BPP}^{\mathsf{PP}}CorrBQP=AdMajBQP=BPPPP and MajBQP=PPP\mathsf{MajBQP} = \mathsf{P}^{\mathsf{PP}}MajBQP=PPP. In fact, we show that other metaphysical modifications of BQP\mathsf{BQP}BQP, such as CBQP\mathsf{CBQP}CBQP (i.e. BQP\mathsf{BQP}BQP with the ability to clone arbitrary quantum states), are also equal to BPPPP\mathsf{BPP}^{\mathsf{PP}}BPPPP. We show that CorrBQP\mathsf{CorrBQP}CorrBQP and MajBQP\mathsf{MajBQP}MajBQP are self-low with respect to classically-accessible queries. In contrast, if they were self-low under quantumly-accessible queries, the counting hierarchy would collapse. Furthermore, we introduce a variant of rational degree that lower-bounds the query complexity of BPPPP\mathsf{BPP}^{\mathsf{PP}}BPPPP. Lastly, we extend the adversary lower-bounding technique to AdPDQP\mathsf{AdPDQP}AdPDQP, BQP\mathsf{BQP}BQP with the ability to sample the current state of an algorithm with collapsing it and adapt the computation based on the samples.
BQP⊆IP\mathsf{BQP} \subseteq \mathsf{IP}BQP⊆IP Does Not Relativize
We construct an oracle relative to which BQP⊈IP\mathsf{BQP} \not\subseteq \mathsf{IP}BQP⊆IP, resolving a long-standing open question in quantum complexity theory. Together with recent work due to Aaronson et al., our work also gives the first oracle separation between IP\mathsf{IP}IP and MIP\mathsf{MIP}MIP, answering a question dating back to Fortnow's thesis. Our separation is based on the Forrelation problem, where given Boolean functions fff and ggg, the goal is to determine if fff is correlated with the Fourier spectrum of ggg. While this task is solvable by a query-efficient quantum algorithm, we show that it admits no classical interactive protocol with polynomial communication and a polynomial-query verifier. Our proof is based on (i) a new structural result showing how to approximate Avg-Max circuits (which are well-known to capture the power of interactive proofs in the oracular setting) by convex functions with small first and second derivatives and (ii) a novel analysis establishing that the Forrelation distribution suggested by Aaronson and Ambainis fools such functions. Our results imply that any prover-efficient classical interactive protocol for BQP\mathsf{BQP}BQP must rely on non-relativizing techniques. This might serve as a partial explanation for the lack of progress towards doubly-efficient, unconditionally sound classical verification of quantum computation.
We study classical query algorithms with post-selection, and find that they are closely connected to rational functions with nonnegative coefficients. We show that the post-selected classical query complexity of a Boolean function is equal to the minimal degree of a rational function with nonnegative coefficients that approximates it (up to a factor of two). For post-selected quantum query algorithms, a similar relationship was shown by Mahadev and de Wolf, where the rational approximations are allowed to have negative coefficients. Using our characterisation, we find an exponentially large separation between post-selected classical query complexity and post-selected quantum query complexity, by proving a lower bound on the degree of rational approximations (with nonnegative coefficients) to the Majority function. This lower bound can be generalised to arbitrary symmetric functions, and allows us to find an unbounded separation between non-deterministic quantum and post-selected classical query complexity. All lower bounds carry over into the communication complexity setting. We show that the zero-error variants of post-selected query algorithms are equivalent to non-deterministic classical query algorithms, which in turn are characterised by nonnegative polynomials, and for some problems require exponentially more queries to the input than their bounded-error counterparts. Finally, we describe a post-selected query algorithm for approximating the Majority function, and an efficient query algorithm for approximate counting.