Phantom Parallelism: Theory & Applications

• Phantom Parallelism is a phenomenon where systems mimic parallel behavior through equivalent serial constructions, revealing apparent speedups.
• It manifests in distinct fields: queueing theory uses the P = FS theorem, modified teleparallel gravity generates phantom-like cosmology, and quantum codes achieve compile-time entangling.
• The concept provides diagnostic criteria and performance benchmarks to distinguish true concurrency from effective, reinterpreted serial processes.

Searching arXiv for the exact term and closely related uses to ground the article in the literature. Phantom parallelism denotes a class of phenomena in which a system exhibits effects ordinarily associated with parallelism, concurrency, or phantom phases, while the operative mechanism is instead an equivalent serial construction, a geometric effective sector, or a compilation-only relabeling. The term appears in distinct technical literatures with different formal content. In queueing theory, it refers to the theorem that a homogeneous array of independent parallel M/M/1 queues is mean-latency equivalent to a tandem of faster serial queues, summarized as $P = FS$ (“parallel is just fast serial”) (Gunther, 2020). In modified teleparallel gravity, it denotes phantom-like cosmological expansion generated by the torsion sector of $f(T)$ gravity without a fundamental phantom scalar field (Karimzadeh et al., 2019). In fault-tolerant quantum computing, it denotes “phantom parallelism,” the ability to realize dense layers of in-block logical entangling gates entirely by compile-time permutations of physical qubits, with no runtime gate, depth, or fidelity cost (Koh et al., 28 Jan 2026). The common thread is an operational indistinguishability at a chosen descriptive level, together with sharp limits on what is and is not preserved.

1. Queueing-theoretic formulation: $P = FS$

In the queueing-theory usage, phantom parallelism is the performance-equivalence phenomenon proven for a class of parallel systems composed of homogeneous, independent single-server queues sharing an aggregate Poisson arrival stream (Gunther, 2020). The paper considers an open network with aggregate Poisson arrivals of rate $\lambda$, $m$ homogeneous independent single-server queues, FCFS discipline, exponential service times with identical mean $S = 1/\mu$, equal routing probabilities, and no synchronization, communication overhead, or shared bottleneck beyond the modeled servers.

Under these assumptions, two subsystems are compared. The parallel subsystem $P$ consists of $m$ independent M/M/1 queues in parallel, each seeing rate $\lambda/m$ and mean service time $S$. The fast serial subsystem $f(T)$0 consists of a tandem chain of $f(T)$1 M/M/1 queues, each seeing arrival rate $f(T)$2 and mean service time $f(T)$3, equivalently service rate $f(T)$4. The theorem states that the mean response time is identical:

$f(T)$5

The proof uses the standard M/M/1 response formula. In $f(T)$6, each job visits exactly one queue with arrival rate $f(T)$7, giving

$f(T)$8

In $f(T)$9, the job visits all $P = FS$0 stages; each stage has response

$P = FS$1

and summing over $P = FS$2 stages yields

$P = FS$3

The stability condition is the same in both representations, $P = FS$4, and the saturation throughput is likewise $P = FS$5 (Gunther, 2020).

The intuition given in the paper is that shorter queues at each parallel processor offset the lack of true concurrency in the equivalent serial construction. At the level of mean response time, capacity, and steady-state completion rate $P = FS$6 for $P = FS$7, the two systems are indistinguishable. This is the sense in which the apparent benefit of parallelism may be “phantom”: the measured speedup can be explained entirely by an equivalent fast-serial model.

2. Scope, non-equivalences, and breakdown conditions in queueing models

The $P = FS$8 theorem is exact only at the mean-response level for the specific constructions above; it is not an equivalence between a parallel array of independent queues and a pooled multiserver queue (Gunther, 2020). The paper explicitly distinguishes the theorem from the M/M/$P = FS$9 model. For a single pooled M/M/$\lambda$0 system, with $\lambda$1 and $\lambda$2, the Erlang C waiting probability is

$\lambda$3

the mean waiting time is

$\lambda$4

and the mean response time is

$\lambda$5

The paper’s conclusion is categorical: the $\lambda$6 theorem does not equate either the parallel M/M/1 array or the fast serial tandem with a single M/M/$\lambda$7 pooling model, and one should not substitute $\lambda$8 into an M/M/1 formula unless one intends to model a genuine single-server system (Gunther, 2020).

A numerical example with $\lambda$9, $m$0, $m$1, and $m$2 illustrates this distinction. The independent parallel queues and the fast serial tandem both yield $m$3, whereas the pooled M/M/4 system yields $m$4 (Gunther, 2020). This difference is central to the interpretation of phantom parallelism: matching the curve $m$5 indicates load splitting across independent queues, not true multiserver pooling.

The theorem preserves mean response time, saturation throughput, and steady-state completion rate, but not queue length distributions, waiting-time distributions beyond the mean, tail latency, variance, transient behavior, synchronization delays, or effects of shared-resource contention and communication overheads (Gunther, 2020). The paper also lists explicit breakdown conditions: unequal splitting without corresponding stage scaling, non-Poisson arrivals such as bursty processes, and shared bottlenecks across parallel servers invalidate the equivalence. For non-exponential service, the Pollaczek–Khinchine formula for M/G/1,

$m$6

suggests that an extension may be possible only at the level of expected mean response and only with careful handling of $m$7; the homogeneous exponential case is the one proven in the paper (Gunther, 2020).

3. Reverse use of $m$8: detection and optimization

The same paper formulates a reverse proposition: given $m$9 tandem queues each with service time $S = 1/\mu$0, reconfiguring them as $S = 1/\mu$1 parallel queues, each still with service time $S = 1/\mu$2, reduces mean response time by a factor of $S = 1/\mu$3, so that $S = 1/\mu$4 (Gunther, 2020). This makes the theorem not only descriptive but diagnostic.

Several empirical tests are proposed for detecting phantom parallelism in measured systems. One test starts from measured speedup $S = 1/\mu$5, where $S = 1/\mu$6 is the mean response under a purported parallelization, and computes the implied effective single-server rate

$S = 1/\mu$7

If $S = 1/\mu$8, or more generally matches the fast-serial mapping, the observed speedup is consistent with phantom parallelism rather than irreducible concurrency (Gunther, 2020). A complementary test infers service time from throughput and utilization: for homogeneous servers, $S = 1/\mu$9, hence $P$0 and $P$1. If the measured per-server utilization follows the predicted split and system latency matches $P$2, the improvement is again consistent with the theorem.

The paper further recommends queueing-signature comparison. If measured $P$3 tracks $P$4, the system behaves like an independent parallel array; if it tracks $P$5, it behaves like the fast tandem representation; if it follows the Erlang C expression $P$6, it behaves like a pooled M/M/$P$7 queue (Gunther, 2020). Tail behavior is also diagnostic: broader tail distributions per queue are more consistent with independent M/M/1 servers than with pooling.

A heterogeneous extension is also developed. For $P$8 parallel queues with service times $P$9 and routing probabilities $m$0 satisfying $m$1, the mean response is

$m$2

The optimal mean response is

$m$3

with $m$4, meaning more traffic is sent to faster servers (Gunther, 2020). Defining effective serial stage times $m$5 produces a fast-serial equivalent with

$m$6

The corresponding capacity is $m$7, assuming independence. For a dual heterogeneous array, the paper gives a numerical example with $m$8, $m$9, and $\lambda/m$0, yielding $\lambda/m$1 and $\lambda/m$2, reproduced exactly by the serial mapping (Gunther, 2020).

The paper also places the result in relation to Amdahl’s and Gustafson’s laws. Replacing service demand $\lambda/m$3 by an effective serial demand $\lambda/m$4 yields an M/M/1-like response

$\lambda/m$5

so that some observed “parallel” speedups can be interpreted as service-time reduction rather than evidence of pooling or stronger stochastic effects (Gunther, 2020).

4. Phantom-like cosmology in modified teleparallel gravity

A distinct usage appears in modified teleparallel gravity, where “phantom parallelism” denotes the realization of effective phantom-like behavior through the geometric torsion sector of $\lambda/m$6 gravity rather than through a fundamental phantom field (Karimzadeh et al., 2019). The cosmological motivation is the Planck 2018 indication that the present-day effective equation-of-state parameter is slightly below $\lambda/m$7, with $\lambda/m$8 (Karimzadeh et al., 2019).

Teleparallel Equivalent of General Relativity replaces curvature by torsion via the Weitzenböck connection, using tetrads $\lambda/m$9 satisfying

$S$0

In the $S$1 extension, the action is

$S$2

where $S$3 and, for a spatially flat FRW background with tetrad $S$4, the torsion scalar reduces to

$S$5

The modified Friedmann equations are written as

$S$6

$S$7

Writing $S$8 permits a GR-like effective-fluid formulation:

$S$9

$f(T)$00

with

$f(T)$01

$f(T)$02

and

$f(T)$03

These equations show that $f(T)$04 can occur even with $f(T)$05, provided the correction term is sufficiently negative (Karimzadeh et al., 2019). In this sense, the torsion sector acts as an effective phantom fluid, but without introducing a scalar with a wrong-sign kinetic term.

The paper emphasizes the contrast with genuine phantom fields. A phantom scalar can produce $f(T)$06 but typically suffers from instabilities, violation of the null energy condition, and quantum-level pathologies. In $f(T)$07 gravity, the effective NEC violation is confined to the geometric sector; no fundamental ghostly field is introduced, and the field equations remain second order, unlike the fourth-order equations of $f(T)$08 gravity (Karimzadeh et al., 2019). The formulation used is the pure-tetrad formulation with vanishing spin connection, which breaks local Lorentz invariance, although covariant formulations can restore it.

5. Model realizations of phantom-like behavior in $f(T)$09 gravity

The paper studies three observationally viable $f(T)$10 ansätze calibrated to $f(T)$11 and $f(T)$12, with a two-parameter scale factor characterized by $f(T)$13 (Karimzadeh et al., 2019). In each case, the criterion for “phantom parallelism” is an effective torsion fluid with $f(T)$14 and a positive, increasing $f(T)$15.

The three models and their reported properties are summarized below.

Model	Form	Reported regime
Power-law	$f(T)$16	$f(T)$17 gives $f(T)$18; crossing occurs in the disfavored direction
Exponential	$f(T)$19	$f(T)$20 gives $f(T)$21; always phantom-like, with no crossing
Combined	$f(T)$22	$f(T)$23 gives $f(T)$24 and crossing at $f(T)$25

For the power-law model, the present-day Friedmann equation fixes

$f(T)$26

with

$f(T)$27

and

$f(T)$28

With $f(T)$29, the model yields $f(T)$30 and positive, increasing $f(T)$31 toward $f(T)$32, but the crossing of the phantom divide is from phantom-like to quintessence-like in the future, which the paper regards as observationally disfavored (Karimzadeh et al., 2019).

For the exponential model,

$f(T)$33

with

$f(T)$34

The sign of $f(T)$35 controls the phase: $f(T)$36 gives $f(T)$37, while $f(T)$38 gives $f(T)$39. A representative value $f(T)$40 produces $f(T)$41 and increasing $f(T)$42, but $f(T)$43 remains below $f(T)$44 at all times, with no crossing of the phantom divide (Karimzadeh et al., 2019).

For the combined model,

$f(T)$45

with

$f(T)$46

At $f(T)$47, the model yields a positive $f(T)$48 that grows toward $f(T)$49, a present-day value $f(T)$50, and a crossing of the phantom divide at $f(T)$51 in the observationally favored direction, from $f(T)$52 at higher redshift to $f(T)$53 near the present (Karimzadeh et al., 2019). Among the three studied models, the paper identifies this combined model as the most cosmologically viable.

The paper also notes that the speed of gravitational waves remains $f(T)$54 in $f(T)$55 models and that current constraints from GW170817/GRB170817A are satisfied. A plausible implication is that the term “phantom” in this literature refers to effective background behavior rather than to the microscopic field content. The author’s conclusion is therefore narrower than a claim of generic stability: viable parameter regions exist, but detailed perturbation studies and broader data analyses remain necessary (Karimzadeh et al., 2019).

6. Quantum-information usage: phantom codes and compilation-level entangling parallelism

In fault-tolerant quantum computing, phantom parallelism is defined within the theory of phantom codes, a class of CSS stabilizer codes in which every in-block logical CNOT is realized by a permutation of physical qubits and therefore absorbed entirely into compilation (Koh et al., 28 Jan 2026). A CSS stabilizer code with parameters $f(T)$56 is phantom if, for every ordered pair $f(T)$57 of distinct logical qubits, there exists a permutation $f(T)$58 of the $f(T)$59 physical qubits such that the induced logical action is the in-block CNOT from $f(T)$60 to $f(T)$61:

$f(T)$62

No physical two-qubit gate or measurement is executed for these in-block logical entanglers. “No spatial or temporal overhead” means that the physical gate count and depth on hardware are unchanged; the logical CNOT layer disappears into a renaming of qubit indices (Koh et al., 28 Jan 2026).

The formal condition is expressed in binary half-symplectic CSS language. Let $f(T)$63 be the X- and Z-type stabilizer generators, $f(T)$64 the X- and Z-type logical generators with $f(T)$65, and $f(T)$66 the permutation matrix of $f(T)$67. A permutation implements a target logical CNOT circuit $f(T)$68 iff

$f(T)$69

together with stabilizer-preservation and orthogonality constraints

$f(T)$70

$f(T)$71

$f(T)$72

The code is phantom if such permutations exist for the full gate set $f(T)$73 (Koh et al., 28 Jan 2026).

“Phantom Parallelism” is then the ability to apply any number of in-block logical entangling gates simultaneously and at perfect fidelity by compilation-only relabeling. Because permutations commute through later circuit layers without operator spread, dense patterns of in-block entanglers compile away into zero-depth, zero-error logical entangling layers within each block (Koh et al., 28 Jan 2026). Interblock entangling gates are not free in general, but phantom in-block entanglers combine with transversal interblock CNOTs. The paper proves that for $f(T)$74 codeblocks, any logical CNOT circuit among those codeblocks can be implemented in physical depth at most $f(T)$75 using transversal CNOTs, up to a residual permutation of logical labels; if all CNOTs are unidirectional, the bound is $f(T)$76 (Koh et al., 28 Jan 2026).

The compiler transformation is purely algebraic. In-block CNOT sublayers are replaced by composed permutations; subsequent single-qubit, fold-diagonal, and transversal interblock CNOT layers are relabeled accordingly; accumulated permutations are pushed to the end and usually dropped rather than physically enacted (Koh et al., 28 Jan 2026). This is the quantum-computing sense in which parallel logical entanglement is “phantom”: the entangling structure is present at the logical level but absent as runtime work.

7. Enumeration, constructions, performance, and limitations of phantom codes

The paper presents an extensive existence theory for phantom codes (Koh et al., 28 Jan 2026). It reports exhaustive enumeration of all $f(T)$77 inequivalent CSS stabilizer codes up to $f(T)$78, modulo qubit permutations and global Hadamards, and identifies $f(T)$79 CSS phantom codes with $f(T)$80 for $f(T)$81, approximately one in $f(T)$82 CSS codes. SAT-based discovery extends the search to $f(T)$83 and yields minimal blocklengths for several $f(T)$84 pairs, including for $f(T)$85 the minimal $f(T)$86 values $f(T)$87 for $f(T)$88, respectively (Koh et al., 28 Jan 2026).

The smallest phantom code is the $f(T)$89 code with stabilizers $f(T)$90 and $f(T)$91. In one logical basis,

$f(T)$92

$f(T)$93

The permutation $f(T)$94 implements $f(T)$95, while $f(T)$96 implements $f(T)$97 (Koh et al., 28 Jan 2026).

Two major families are then constructed. The qRM-based family begins from quantum Reed–Muller CSS codes and promotes selected degree-$f(T)$98 logicals to stabilizers, yielding phantom codes with

$f(T)$99

Representative examples include $P = FS$00, $P = FS$01, $P = FS$02, and $P = FS$03 (Koh et al., 28 Jan 2026). A second family is obtained by binarization and concatenation from GF(4) codes, producing $P = FS$04 phantom codes of length $P = FS$05 and distance at least $P = FS$06, with explicit instances $P = FS$07, $P = FS$08, $P = FS$09, $P = FS$10, and $P = FS$11 (Koh et al., 28 Jan 2026). Additional constructions include punctured hypercube codes and hypergraph-product constructions.

The gate set is constrained by a no-go theorem: if a stabilizer code implements a logical gate $P = FS$12 by qubit permutations, then no strictly transversal $P = FS$13 on any number of blocks with $P = FS$14 can exist. For phantom codes this excludes strictly transversal $P = FS$15, except degenerate commuting variants (Koh et al., 28 Jan 2026). Nevertheless, the paper identifies diagonal fold gates such as $P = FS$16 and $P = FS$17, teleported Hadamards, and certain decoupled non-Clifford constructions.

The performance claims are based on end-to-end noisy simulations with state preparation, full QEC cycles, and neutral-atom-calibrated circuit-level noise. The reported error model uses single-qubit gate depolarizing error $P = FS$18, two-qubit gate depolarizing error $P = FS$19, idle depolarizing error $P = FS$20, measurement bit-flip error $P = FS$21, and reset $P = FS$22 flips $P = FS$23, with results at $P = FS$24 and $P = FS$25 (Koh et al., 28 Jan 2026). For the $P = FS$26 qRM phantom code, strict preselection acceptance is approximately $P = FS$27 at $P = FS$28 and approximately $P = FS$29 at $P = FS$30; relaxed acceptance is approximately $P = FS$31 and approximately $P = FS$32, respectively (Koh et al., 28 Jan 2026).

On single-block repeated in-block CNOT tasks, the paper reports that the phantom code’s failure rate remains flat as the number of repeated in-block CNOT layers increases, because those layers are free. At $P = FS$33, after state preparation only, the $P = FS$34 phantom code with strict preselection achieves approximately $P = FS$35 lower failure rate than a surface code at $P = FS$36 with similar footprint and approximately $P = FS$37 lower than $P = FS$38. By $P = FS$39 repeated in-block CNOT layers, it is approximately $P = FS$40 better than $P = FS$41 and approximately $P = FS$42 better than $P = FS$43 (Koh et al., 28 Jan 2026). For logical GHZ preparation up to $P = FS$44, the phantom code attains approximately $P = FS$45 lower infidelity than the $P = FS$46 surface code at comparable qubit counts; for Trotterized many-body simulation at $P = FS$47, it achieves approximately $P = FS$48 lower infidelity than $P = FS$49 and approximately $P = FS$50 better than $P = FS$51 at nearly identical footprint (Koh et al., 28 Jan 2026).

The paper is explicit about limitations. Known phantom codes are non-LDPC and require Steane-style QEC with verified ancillas; state-preparation factories with preselection introduce acceptance overhead; current error-correcting families have encoding rate $P = FS$52; and non-Clifford gates available through current constructions are effectively distance-2 (Koh et al., 28 Jan 2026). Hardware integration is also nontrivial, because some runtime stacks may still enforce data motion despite the compile-time nature of the permutations.

Across these three literatures, phantom parallelism therefore names different but structurally related ideas: mean-level equivalence between parallel and fast serial queueing systems (Gunther, 2020), effective phantom behavior sourced by geometry rather than ghost fields in $P = FS$53 cosmology (Karimzadeh et al., 2019), and logical entangling parallelism compiled away into permutations in fault-tolerant quantum codes (Koh et al., 28 Jan 2026). In each case, the central scientific content lies not in a generic metaphor of “apparent parallelism,” but in a formal equivalence or effective description with clearly delimited invariants, breakdown conditions, and practical consequences.