Constrained Postselection Theorem

• Constrained Postselection is defined by restricting conditioning to specific events or linear functions, enabling exact characterizations in quantum query complexity, Bell tests, and time-space models.
• It establishes a precise equivalence between postselection query complexity and approximate rational degree, showing that rational approximations can be derived directly from constrained quantum algorithms.
• The framework also demonstrates loophole-free postselection in Bell experiments and equates postselection with restart in bounded-time and space machines, with implications for quantum proof systems.

Constrained postselection denotes, in several strands of quantum information and complexity theory, a setting in which conditioning on a designated event is permitted only under explicit structural restrictions, and the restricted model then admits an exact characterization. In quantum query complexity, postselection query complexity coincides with approximate rational degree up to a factor of $2$; in Bell scenarios, linear post-selection is loophole-free for local hidden-variable theories; in time-space-bounded machine models, postselection is equivalent to restart; and in two-prover quantum proof systems, postselection converts exponentially small-gap protocols into a characterization of $\mathrm{NEXP}$. A related use appears in work on adaptive measurements, where the correspondence with postselection is argued informally rather than established as a standard theorem (Mahadev et al., 2014, Hoban et al., 2011, Yakaryilmaz et al., 2011, Kinoshita, 2018, Dhillon, 2011).

1. Scope and recurring structure

The literature summarized here uses the expression in several related senses rather than for a single canonical statement. Each formulation restricts postselection in a different way and then identifies the exact power of the resulting model.

Setting	Constraint	Main characterization
Quantum query complexity	Designated postselection bit $a$, with $\Pr[a=1]>0$, conditioning only on $a=1$	$$\operatorname{PostQ}_\varepsilon(f)\le \operatorname{rdeg}_\varepsilon(f)\le 2\,\operatorname{PostQ}_\varepsilon(f)$$ (Mahadev et al., 2014)
Multi-party Bell tests	$s_j=g_j(x)$ or $s_j=g_j(m^{\setminus j},x)$ with all $g_j$ linear	LHV correlators remain the convex hull of linear Boolean functions (Hoban et al., 2011)
PTMs and QTMs under time and space bounds	Same time and space resources in both models	Postselection is identical to restart (Yakaryilmaz et al., 2011)
Two-prover quantum proof systems	Perfect completeness and exponentially small completeness–soundness gap	$\mathrm{postQMA}(2)(1,s)=\mathrm{NEXP}$ for some constant $\mathrm{NEXP}$0 (Kinoshita, 2018)
Adaptive measurements	Simulation argument via MBQC/circuit equivalence and $\mathrm{NEXP}$1	Conceptual claim, not a rigorous standard theorem (Dhillon, 2011)

This suggests that the adjective “constrained” does not single out one formal template. Instead, it marks a recurring research program: determine which restrictions on postselection preserve a tractable structure, and then characterize exactly what the restricted resource computes, approximates, or certifies.

2. Query-complexity formulation and rational approximation

In the query model, the central statement is that quantum query algorithms with postselection are exactly the algorithmic counterpart of low-degree rational approximations (Mahadev et al., 2014). The starting point is the Beals et al. polynomial method: a standard quantum query algorithm with $\mathrm{NEXP}$2 queries has acceptance probability equal to a polynomial of degree at most $\mathrm{NEXP}$3 in the input bits. Postselection changes that polynomial structure into a rational one.

A postselection algorithm is a quantum query algorithm with two output bits $\mathrm{NEXP}$4, and it computes $\mathrm{NEXP}$5 with error $\mathrm{NEXP}$6 if for every input $\mathrm{NEXP}$7,

$\mathrm{NEXP}$8

The postselection query complexity $\mathrm{NEXP}$9 is the minimum number of queries among such algorithms.

The parallel approximation-theoretic notion is rational approximation. For $a$0, a rational function $a$1 with $a$2 everywhere has degree

$a$3

It $a$4-approximates $a$5 if

$a$6

and the minimum such degree is the $a$7-approximate rational degree $a$8.

The theorem pair proved in this framework is

$a$9

and

$\Pr[a=1]>0$0

for all $\Pr[a=1]>0$1 and Boolean $\Pr[a=1]>0$2. Hence

$\Pr[a=1]>0$3

The forward implication is immediate from the query polynomial method. If a postselection algorithm uses $\Pr[a=1]>0$4 queries, then $\Pr[a=1]>0$5 and $\Pr[a=1]>0$6 are polynomials of degree at most $\Pr[a=1]>0$7. Their ratio,

$\Pr[a=1]>0$8

is therefore a rational function of degree at most $\Pr[a=1]>0$9, and the postselection condition ensures $a=1$0-approximation of $a=1$1.

The reverse implication is the nontrivial direction. Given a degree-$a=1$2 rational approximation $a=1$3, the construction uses Fourier decompositions of $a=1$4 and $a=1$5, prepares a superposition encoding the Fourier coefficients, queries the input to attach phases $a=1$6, applies a Hadamard transform, and postselects on the all-zero outcome in one register. The residual qubit depends on $a=1$7 and

$a=1$8

After a final Hadamard and measurement, the sign of $a=1$9 determines $$\operatorname{PostQ}_\varepsilon(f)\le \operatorname{rdeg}_\varepsilon(f)\le 2\,\operatorname{PostQ}_\varepsilon(f)$$0. In this formulation, postselection is “constrained” because there is a designated postselection bit $$\operatorname{PostQ}_\varepsilon(f)\le \operatorname{rdeg}_\varepsilon(f)\le 2\,\operatorname{PostQ}_\varepsilon(f)$$1, the condition $$\operatorname{PostQ}_\varepsilon(f)\le \operatorname{rdeg}_\varepsilon(f)\le 2\,\operatorname{PostQ}_\varepsilon(f)$$2 is required on every input, and conditioning is only on the event $$\operatorname{PostQ}_\varepsilon(f)\le \operatorname{rdeg}_\varepsilon(f)\le 2\,\operatorname{PostQ}_\varepsilon(f)$$3. The paper emphasizes that this is weaker than arbitrary magical conditioning, yet strong enough to match rational approximation complexity exactly up to a factor of $$\operatorname{PostQ}_\varepsilon(f)\le \operatorname{rdeg}_\varepsilon(f)\le 2\,\operatorname{PostQ}_\varepsilon(f)$$4.

3. Majority, optimality, and Newman’s theorem

The query-theoretic theorem becomes especially concrete for the Majority function

$$\operatorname{PostQ}_\varepsilon(f)\le \operatorname{rdeg}_\varepsilon(f)\le 2\,\operatorname{PostQ}_\varepsilon(f)$$5

where the postselection framework yields an optimal algorithm up to constant factors (Mahadev et al., 2014).

Earlier work of Aaronson had shown that Majority can be computed with postselection in polylogarithmic queries, with

$$\operatorname{PostQ}_\varepsilon(f)\le \operatorname{rdeg}_\varepsilon(f)\le 2\,\operatorname{PostQ}_\varepsilon(f)$$6

and the paper also cites an $$\operatorname{PostQ}_\varepsilon(f)\le \operatorname{rdeg}_\varepsilon(f)\le 2\,\operatorname{PostQ}_\varepsilon(f)$$7-query constant-error version from earlier work. The improvement proved here is

$$\operatorname{PostQ}_\varepsilon(f)\le \operatorname{rdeg}_\varepsilon(f)\le 2\,\operatorname{PostQ}_\varepsilon(f)$$8

The construction is built from two auxiliary postselected procedures. The first, an Aaronson-style one-query gadget, prepares a qubit proportional to

$$\operatorname{PostQ}_\varepsilon(f)\le \operatorname{rdeg}_\varepsilon(f)\le 2\,\operatorname{PostQ}_\varepsilon(f)$$9

for $s_j=g_j(x)$0 with $s_j=g_j(x)$1. This encodes the Hamming weight into amplitudes. The algorithm then varies parameters and checks closeness to $s_j=g_j(x)$2. The second and third components are the two distinguishing procedures: $s_j=g_j(x)$3 queries suffice to distinguish

$s_j=g_j(x)$4

with success probability at least $s_j=g_j(x)$5, and

$s_j=g_j(x)$6

queries suffice to distinguish

$s_j=g_j(x)$7

with success probability $s_j=g_j(x)$8. Setting

$s_j=g_j(x)$9

and combining the two procedures yields the stated bound.

The same theorem has an approximation-theoretic consequence. Sherstov proved the lower bound

$s_j=g_j(m^{\setminus j},x)$0

on the $s_j=g_j(m^{\setminus j},x)$1-approximate rational degree of $s_j=g_j(m^{\setminus j},x)$2. By the postselection–rational equivalence, this lower bound transfers to $s_j=g_j(m^{\setminus j},x)$3, so the Majority algorithm is optimal up to a constant factor.

The paper then derives Newman’s theorem from the Majority construction. First, it obtains a degree-$s_j=g_j(m^{\setminus j},x)$4 rational approximation $s_j=g_j(m^{\setminus j},x)$5 to $s_j=g_j(m^{\setminus j},x)$6 on $s_j=g_j(m^{\setminus j},x)$7 with

$s_j=g_j(m^{\setminus j},x)$8

and with

$s_j=g_j(m^{\setminus j},x)$9

Second, multiplying by $g_j$0 yields an approximation to $g_j$1, giving the corollary that for every $g_j$2, there exists a degree-$g_j$3 rational function $g_j$4 such that

$g_j$5

In this setting, the constrained postselection theorem does more than identify a complexity measure: it turns postselected query constructions into rational approximation theorems.

4. Loophole-free post-selection in Bell scenarios

A distinct formulation appears in multi-party CHSH-type Bell experiments, where the theorem identifies exactly which post-selection rules preserve the local hidden-variable region (Hoban et al., 2011). There are $g_j$6 spatially separated parties, each with a binary setting

$g_j$7

and a binary outcome

$g_j$8

The object of study is the parity correlator

$g_j$9

Under space-like separation and measurement independence, the paper’s Theorem 1 states that the LHV correlators are exactly the convex hull of deterministic linear Boolean functions of the settings. Deterministically,

$\mathrm{postQMA}(2)(1,s)=\mathrm{NEXP}$0

so that

$\mathrm{postQMA}(2)(1,s)=\mathrm{NEXP}$1

which is linear in $\mathrm{postQMA}(2)(1,s)=\mathrm{NEXP}$2. The LHV polytope is therefore the convex hull of linear Boolean functions, with Bell inequalities as its facets.

The paper then isolates post-selection rules that do not enlarge this polytope. In setting post-selection (SP), one fixes

$\mathrm{postQMA}(2)(1,s)=\mathrm{NEXP}$3

Theorem 2 states that the correlators

$\mathrm{postQMA}(2)(1,s)=\mathrm{NEXP}$4

obtained after post-selecting on $\mathrm{postQMA}(2)(1,s)=\mathrm{NEXP}$5 lie in the convex hull of linear $\mathrm{postQMA}(2)(1,s)=\mathrm{NEXP}$6 iff all $\mathrm{postQMA}(2)(1,s)=\mathrm{NEXP}$7 are linear. In setting-output post-selection (SOP), the settings may depend on both the conditioning variable and other outcomes,

$\mathrm{postQMA}(2)(1,s)=\mathrm{NEXP}$8

Theorem 3 states that the LHV correlators remain in the convex hull of linear $\mathrm{postQMA}(2)(1,s)=\mathrm{NEXP}$9 iff all $\mathrm{NEXP}$00 are linear.

The contrast with ordinary post-selection is illustrated by the example

$\mathrm{NEXP}$01

where $\mathrm{NEXP}$02 is random. Post-selecting on $\mathrm{NEXP}$03 effectively maps $\mathrm{NEXP}$04 into $\mathrm{NEXP}$05, and the remaining correlator becomes

$\mathrm{NEXP}$06

which is non-linear. This is the detection loophole: post-selection changes the sample in a way that can correlate settings with hidden variables, allowing LHV theories to fake Bell violations.

The constrained theorem in this context states that linear SP and linear SOP are loophole-free: they preserve the LHV polytope exactly. At the same time, quantum correlators can become strictly richer under the same linear constraints. Theorem 4 shows that for $\mathrm{NEXP}$07 parties and $\mathrm{NEXP}$08, the correlator

$\mathrm{NEXP}$09

can be realized using linear SOP, but not with SP alone. The construction uses two GHZ states and adaptive measurement choices. The paper further notes that there is no analogous enhancement in the bipartite case $\mathrm{NEXP}$10. In this formulation, “constrained postselection” identifies the largest natural class of post-selection rules that preserve the classical boundary while still enlarging the quantum region.

5. Time-space-bounded machines and equivalence with restart

In machine-based complexity theory, the constrained postselection theorem states that postselection is equivalent to restart when time and space bounds are matched exactly (Yakaryilmaz et al., 2011). A machine with postselection partitions its state set into continuing states $\mathrm{NEXP}$11, postselection accept states $\mathrm{NEXP}$12, postselection reject states $\mathrm{NEXP}$13, and nonpostselection halting states $\mathrm{NEXP}$14, with the requirement that for every input the machine reaches $\mathrm{NEXP}$15 with nonzero probability. A machine with restart has an extra outcome $\mathrm{NEXP}$16 and a set of restarting states $\mathrm{NEXP}$17, so that each round ends in accept, reject, or restart.

The central theorem is: for any time bound $\mathrm{NEXP}$18 and space bound $\mathrm{NEXP}$19, the class of languages recognized by $\mathrm{NEXP}$20-time and $\mathrm{NEXP}$21-space PTMs or QTMs with postselection is identical to the class of languages recognized by $\mathrm{NEXP}$22-time PTMs or QTMs with restart using space $\mathrm{NEXP}$23. The same equality is also valid for the real-time, in particular, finite memory, versions of these models. The simulation is direct in both directions. Postselection states become accept/reject states of the restart machine, and nonpostselection halting states become restart states; conversely, restart states become nonpostselection halting states. Because both models compute final acceptance probabilities by the same normalization formula, the recognized language and error bound remain unchanged.

This equivalence immediately implies that for purely space-bounded machines, postselection does not increase power. The paper then shows that the effect changes under simultaneous time and space constraints. In the probabilistic setting, PTMs with postselection that use $\mathrm{NEXP}$24 space and that have polynomial expected runtime are strictly more powerful than their standard versions. The separating witness is

$\mathrm{NEXP}$25

which is recognized by a real-time PTM with restart using $\mathrm{NEXP}$26 space, while standard polynomial-time PTMs using $\mathrm{NEXP}$27 space recognize only the regular languages. The same separation holds for one-way PTMs with postselection that use $\mathrm{NEXP}$28 space.

In the quantum setting, the paper proves

$\mathrm{NEXP}$29

Here the witness language is the nonregular palindrome language

$\mathrm{NEXP}$30

which is recognized by a real-time QTM with restart using $\mathrm{NEXP}$31 space, whereas

$\mathrm{NEXP}$32

Using the strict inclusion

$\mathrm{NEXP}$33

the postselected versions satisfy

$\mathrm{NEXP}$34

The paper also records structural properties. The classes

$\mathrm{NEXP}$35

are closed under complementation, union, and intersection. Exact postselected classes satisfy

$\mathrm{NEXP}$36

For finite automata, the probabilistic one-sided and exact classes collapse,

$\mathrm{NEXP}$37

whereas the quantum classes remain distinct: $\mathrm{NEXP}$38 A notable corollary stated in the paper is that $\mathrm{NEXP}$39 would equal $\mathrm{NEXP}$40 if randomized machines had the postselection capability.

6. Small-gap proof systems, adaptive measurements, and interpretive limits

A further constrained formulation arises in two-prover quantum Merlin–Arthur proof systems with postselection (Kinoshita, 2018). In $\mathrm{NEXP}$41, a polynomial-time verifier receives two unentangled quantum witnesses and accepts with probability exceeding $\mathrm{NEXP}$42 on yes-instances and below $\mathrm{NEXP}$43 on no-instances. In $\mathrm{NEXP}$44, the verifier outputs an ordinary output bit $\mathrm{NEXP}$45 and a postselection bit $\mathrm{NEXP}$46, with the requirement that

$\mathrm{NEXP}$47

for all witnesses, and correctness is evaluated through the conditional probability

$\mathrm{NEXP}$48

The main theorem states that there exists a constant $\mathrm{NEXP}$49 such that

$\mathrm{NEXP}$50

The easy direction is

$\mathrm{NEXP}$51

via exponential-time classical simulation of polynomial-size quantum witnesses and polynomial-size verifier circuits. The hard direction proceeds through two ingredients. First, for any polynomial $\mathrm{NEXP}$52,

$\mathrm{NEXP}$53

Second, there exists a constant $\mathrm{NEXP}$54 such that

$\mathrm{NEXP}$55

The conversion protocol applies $\mathrm{NEXP}$56, copies the output qubit, applies $\mathrm{NEXP}$57, prepares a one-qubit state $\mathrm{NEXP}$58, performs a small rotation, postselects on the subspace

$\mathrm{NEXP}$59

and then measures in the basis

$\mathrm{NEXP}$60

With

$\mathrm{NEXP}$61

the yes-instance case retains perfect completeness, while the no-instance case yields a constant soundness bound after postselection. The paper stresses that earlier postselection proofs for $\mathrm{NEXP}$62 and $\mathrm{NEXP}$63 do not directly transfer, because the $\mathrm{NEXP}$64 setting does not permit the same garbage-erasure or eigenstate-based reductions.

The same paper presents this as a broader principle: for some proof systems with perfect completeness and exponentially small soundness gap, postselection converts a tiny-gap verifier into a constant-gap postselection class. In the $\mathrm{NEXP}$65 case, this identifies the exact power of the postselected model as $\mathrm{NEXP}$66.

A thematically related but methodologically different claim appears in work on adaptive measurements (Dhillon, 2011). There the central statement is that adaptive measurements in a quantum computational model can be simulated by postselected measurements. The argument proceeds informally through three ingredients: MBQC employs adaptive local measurements on a resource state; measurement-based models can be simulated on the quantum circuit model; and

$\mathrm{NEXP}$67

The intended conclusion is that adaptive-measurement computation can be reproduced in a postselected framework, potentially with consequences such as the ability to solve $\mathrm{NEXP}$68 problems. However, the paper explicitly does not provide a rigorous standard complexity-theoretic proof, an explicit gate-by-gate simulation, a quantitative overhead, or a branch-by-branch theorem for arbitrary adaptive measurement trees. Its relation to constrained postselection is therefore conceptual rather than formal.

Taken together, these formulations show that constrained postselection is not a single theorem but a technical pattern. In each case, postselection is restricted by a designated postselection bit, a linear conditioning rule, a matched resource bound, or a small-gap hypothesis; and under that restriction, one obtains an exact theorem rather than an unconstrained appeal to postselection as a generic power amplification device.