Postselected Quantum Query Complexity

• Postselected quantum query complexity is a model that enhances quantum algorithms by discarding undesired outcomes to achieve tight rational approximations of Boolean functions.
• It establishes an equivalence between the rational approximation degree and query complexity, offering insights into adaptive query limitations and oracle separations.
• Applications include optimal postselection algorithms for functions like MAJ, hardness amplification techniques, and demonstrating barriers to collapsing quantum adaptive complexity classes.

Postselected quantum query complexity investigates the power of quantum algorithms in the query (oracle) model when enhanced with the operation of postselection. Postselection refers to the ability to discard computational paths based on a chosen measurement outcome, retaining only those that yield a specific result. This operation fundamentally elevates the computational strength of quantum algorithms, corresponding to a transition from BQP to PP in the circuit model. In the query complexity setting, postselection is tightly linked to rational approximation of Boolean functions, enabling tight characterizations and establishing connections to core areas in quantum complexity theory, including hardness amplification and oracle separations.

1. Formal Definitions

A postselected quantum query algorithm for $f:\{0,1\}^n \to \{0,1\}$ with error $\epsilon\in[0,1/2)$ is a quantum query algorithm making $T$ queries, producing two output bits $a, b \in \{0,1\}$ so that for all $x$:

• $\Pr[a = 1] > 0$, and
• $\Pr[b = f(x)\mid a = 1] \ge 1-\epsilon$.

The minimal $T$ for which such an $\epsilon$-error postselection algorithm for $f$ exists is $\mathrm{PostQ}_\epsilon(f)$. In the adaptive query setting, the algorithm can submit queries that depend adaptively on previous answers.

Rational and polynomial approximations are central. For a Boolean $f$, the $\epsilon$-approximate rational degree $\mathrm{rdeg}_\epsilon(f)$ is the minimal degree $d$ for which there exists a rational function $R$ of degree $d$ such that $|R(x) - f(x)| \le \epsilon$ for all $x\in\{0,1\}^n$ (Mahadev et al., 2014).

The one-sided low-weight approximate degree, $\mathrm{mwdeg}^{1/2}(f)$, minimizes the degree plus the logarithm of the sum of absolute values of coefficients, over polynomials $p$ with $p(x)\ge 1$ if $f(x)=1$ and $p(x)\le 1/2$ if $f(x)=0$ (Chen, 2016).

2. Characterization via Rational Approximation

Postselected quantum query complexity and rational approximation degree are equivalent up to constant factors. For every Boolean function $f$ and $\epsilon < 1/2$: $$\mathrm{rdeg}_\epsilon(f) \le 2\cdot\mathrm{PostQ}_\epsilon(f) \le 2\cdot\mathrm{rdeg}_\epsilon(f)$$ This equivalence is established by constructing rational approximations from postselection algorithms (via polynomialization of measurement probabilities), and conversely, simulating rational approximations by quantum algorithms that use postselection (Mahadev et al., 2014).

In particular, for each $f$ and $\epsilon$,

• A postselection quantum query algorithm with $T$ queries yields a rational approximation of $f$ of degree at most $2T$.
• A rational approximation of degree $d$ yields a quantum postselection algorithm making $d$ queries.

The one-sided low-weight approximate degree $\mathrm{mwdeg}^{1/2}(f)$ further characterizes the PP-analogue of postselected quantum query complexity in the oracle model (Chen, 2016).

3. Limits of Postselection for Adaptive Queries

Adaptive versions of Boolean functions, denoted $\mathrm{Ada}_{f,d}$, recursively encode $d+1$ queries where each is adaptively chosen based on previous outputs, structured as a decision tree of depth $d$.

Main theorems demonstrate that postselection does not efficiently simulate adaptive quantum queries beyond logarithmic depth, assuming the original function $f$ requires super-polylogarithmic query complexity under bounded error:

• If $f$ is hard for bounded-error quantum or classical query algorithms (i.e., requires $\omega(\mathrm{polylog}(n))$ queries), then so is its adaptive version under postselection, e.g., $\mathrm{Ada}_{f,d} \not\in \mathrm{PostBQP}^{dt}$ (Chen, 2016).
• Quantitatively, if $\mathrm{mwdeg}^{1/2}(f) > T$, then $$\mathrm{mwdeg}^{1/2}(\mathrm{Ada}_{f,d}) > \min(T/4,2^{d-1})$$.

This establishes a barrier: postselection does not collapse adaptive query complexity in the quantum (nor classical) case, a relativized counterpart to hierarchy separations in Turing machine complexity.

4. Hardness Amplification and Degree Lower Bounds

Postselected query complexity underpins explicit constructions amplifying polynomial hardness. For $f$ with large one-sided approximate degree, a function $F = \mathrm{Ada}_{f,d}$ is constructed such that $F$ requires high degree to approximate even at error $1/2 - 2^{-m}$ for $m = 2^d$: $\deg^{1/2 - 2^{-m}}(F) \geq \deg^{2/3}(f)$ For $N = (2^{d+1} - 1)n$ inputs, this achieves hardness amplification comparable to other approaches, with only $O(\log n)$-depth adaptive composition (Chen, 2016).

This demonstrates the quantitative inadequacy of postselection in the face of amplified one-sided hardness: the adaptive function remains hard to approximate for postselection-based algorithms.

5. Oracle Separations and Complexity Implications

The limits of postselected quantum query complexity are leveraged to establish new oracle separations among complexity classes for adaptive queries:

• Taking $f(x)=\mathrm{AND}_n$ with $$\mathrm{mwdeg}^{1/2}(\mathrm{AND}_n)=\Omega(\sqrt{n})$$ and $d=\log n$, it follows $$\mathrm{PostQ}^{dt}(\mathrm{Ada}_{\mathrm{AND}_n})=\Omega(\sqrt{n})$$, while $\mathrm{Ada}_{\mathrm{AND}_n}\in P^{NP[O(\log n)]}$. Thus, there exists an oracle $O$ such that $P^{NP^O} \not\subset PP^O$.
• For $f$ the Permutation Testing Problem, analogous reasoning leads to $P^{SZK^O} \not\subset PP^O$.

These results show that adaptive queries can be fundamentally harder than postselection can efficiently resolve, even for quantum algorithms. This suggests limits to relativized collapses to $\mathsf{PP}$ in complexity theory (Chen, 2016).

6. Optimality and Explicit Algorithms

An explicit, essentially optimal postselection quantum algorithm for the Majority function ($\mathrm{MAJ}_n$) is constructed: $$\mathrm{PostQ}_\epsilon(\mathrm{MAJ}_n) = \Theta\left( \log\left(\frac{n}{\log (1/\epsilon)}\right) \cdot \log(1/\epsilon) \right)$$ This matches Sherstov's lower bounds on rational approximation degree up to constants and is achieved by polynomially many logarithmic rounds of "weight elimination" strategies, combining the power of quantum state preparation and postselection to filter Hamming weights (Mahadev et al., 2014).

Additionally, Newman’s theorem for rational approximation of $|z|$ on $[-1,1]$ (with error $2^{-2\sqrt{d}}$ for degree-$d$ rationals) is recovered from these constructions, illustrating how postselection quantum query models naturally yield classical results in approximation theory.

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Summary table: Core Concepts and Correspondence

Concept	Formalization	Reference
Postselected quantum query complexity	$\mathrm{PostQ}_\epsilon(f)$	(Mahadev et al., 2014, Chen, 2016)
Rational approximation degree	$\mathrm{rdeg}_\epsilon(f)$	(Mahadev et al., 2014)
One-sided low-weight approximate degree	$\mathrm{mwdeg}^{1/2}(f)$	(Chen, 2016)
Adaptive function construction	$\mathrm{Ada}_{f,d}$	(Chen, 2016)
Oracle separation	$P^{NP^O}\not\subset PP^O$, $P^{SZK^O}\not\subset PP^O$	(Chen, 2016)
MAJ postselection query complexity	$$\Theta\left( \log\left(\frac{n}{\log (1/\epsilon)}\right) \cdot\log(1/\epsilon)\right)$$	(Mahadev et al., 2014)

7. Connections and Outlook

Postselected quantum query complexity serves as a nexus linking quantum computing, complexity theory, and approximation theory. Its characterization by rational approximation situates query algorithms within the broader context of functional analysis. Limitations on adaptive simulation, proven via polynomial degree methods, inform both the power and inherent barriers of postselected computation models. Oracle separation results reaffirm the distinctiveness of adaptive computation beyond postselection, while explicit algorithmic constructions showcase the potential for transferring quantum techniques into classical approximation and hardness amplification.

Key results in (Mahadev et al., 2014) and (Chen, 2016) demonstrate both the breadth and the limitations of postselection as an algorithmic principle in the quantum query setting, establishing a foundation for future advances in quantum complexity and approximation theory.