# Non-Causal Computation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Logical Consistency

## 3. Non-Causal Circuit Model

## 4. Computational Advantage

## 5. Other Non-Causal Computational Models

## 6. Conclusions and Open Questions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Fredkin, E.; Toffoli, T. Conservative logic. Int. J. Theor. Phys.
**1982**, 21, 219–253. [Google Scholar] [CrossRef] - Echeverria, F.; Klinkhammer, G.; Thorne, K.S. Billiard balls in wormhole spacetimes with closed timelike curves: Classical theory. Phys. Rev. D
**1991**, 44, 1077–1099. [Google Scholar] [CrossRef] - Chiribella, G. Perfect discrimination of no-signalling channels via quantum superposition of causal structures. Phys. Rev. A
**2012**, 86, 040301. [Google Scholar] [CrossRef] - Colnaghi, T.; D’Ariano, G.M.; Facchini, S.; Perinotti, P. Quantum computation with programmable connections between gates. Phys. Lett. A
**2012**, 376, 2940–2943. [Google Scholar] [CrossRef] - Chiribella, G.; D’Ariano, G.M.; Perinotti, P.; Valiron, B. Quantum computations without definite causal structure. Phys. Rev. A
**2013**, 88, 022318. [Google Scholar] [CrossRef] - Araújo, M.; Costa, F.; Brukner, Č. Computational Advantage from Quantum-Controlled Ordering of Gates. Phys. Rev. Lett.
**2014**, 113, 250402. [Google Scholar] [CrossRef] [PubMed] - Procopio, L.M.; Moqanaki, A.; Araújo, M.; Costa, F.; Alonso Calafell, I.; Dowd, E.G.; Hamel, D.R.; Rozema, L.A.; Brukner, Č.; Walther, P. Experimental superposition of orders of quantum gates. Nat. Commun.
**2015**, 6, 7913. [Google Scholar] [CrossRef] [PubMed] - Oreshkov, O.; Costa, F.; Brukner, Č. Quantum correlations with no causal order. Nat. Commun.
**2012**, 3, 1092. [Google Scholar] [CrossRef] [PubMed] - Baumeler, Ä.; Feix, A.; Wolf, S. Maximal incompatibility of locally classical behavior and global causal order in multiparty scenarios. Phys. Rev. A
**2014**, 90, 042106. [Google Scholar] [CrossRef] - Baumeler, Ä.; Wolf, S. The space of logically consistent classical processes without causal order. New J. Phys.
**2016**, 18, 013036. [Google Scholar] [CrossRef] - Branciard, C.; Araújo, M.; Feix, A.; Costa, F.; Brukner, Č. The simplest causal inequalities and their violation. New J. Phys.
**2016**, 18, 013008. [Google Scholar] [CrossRef] - Feix, A.; Araújo, M.; Brukner, Č. Quantum superposition of the order of parties as a communication resource. Phys. Rev. A
**2015**, 92, 052326. [Google Scholar] [CrossRef] - Tegmark, M. The Interpretation of Quantum Mechanics: Many Worlds or Many Words? Fortschr. Phys.
**1998**, 46, 855–862. [Google Scholar] [CrossRef] - Aaronson, S. Guest Column: NP-complete problems and physical reality. ACM SIGACT News
**2005**, 36, 30–52. [Google Scholar] [CrossRef] - Baumeler, Ä.; Wolf, S. Device-independent test of causal order and relations to fixed-points. New J. Phys.
**2016**, 18, 035014. [Google Scholar] [CrossRef] - Everett, H. “Relative State” Formulation of Quantum Mechanics. Rev. Mod. Phys.
**1957**, 29, 454–462. [Google Scholar] [CrossRef] - Bennett, C.H. Logical Reversibility of Computation. IBM J. Res. Dev.
**1973**, 17, 525–532. [Google Scholar] [CrossRef] - Valiant, L.G.; Vazirani, V.V. NP is as easy as detecting unique solutions. Theor. Comput. Sci.
**1986**, 47, 85–93. [Google Scholar] [CrossRef] - Deutsch, D. Quantum mechanics near closed timelike lines. Phys. Rev. D
**1991**, 44, 3197–3217. [Google Scholar] [CrossRef] - Aaronson, S.; Watrous, J. Closed timelike curves make quantum and classical computing equivalent. Proc. R. Soc. A Math. Phys. Eng. Sci.
**2009**, 465, 631–647. [Google Scholar] [CrossRef] - Brun, T.A.; Wilde, M.M.; Winter, A. Quantum State Cloning Using Deutschian Closed Timelike Curves. Phys. Rev. Lett.
**2013**, 111, 190401. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Causal and non-causal computation. The arrows point in the direction of computation. (

**a**) The values that are assigned to the variables of a computational model at time t depend on ${\rho}_{t-1}$. (

**b**) Cyclic dependencies of the values that are assigned to the variables at different steps during the computation.

**Figure 2.**(

**a**) Overdetermined circuit: The bit 0 is mapped to 1 and vice versa; i.e., there is no consistent assignment of a value that travels on the wire. (

**b**) Information antinomy: Both 0 and 1 could potentially travel on the wire, yet the circuit does not specify which.

**Figure 3.**(

**a**) Open circuit $\mathcal{C}$ with input a. (

**b**) Closed circuit ${\mathcal{C}}_{i}$ with $a=i\to {c}_{a}={c}_{i}$. (

**c**) The big box represents a non-causal comb (note that combs obey causality; the higher-order transformations described here are equivalent to combs, yet where the causality assumption is dropped) that transforms a gate (${H}^{\prime}$) to a new gate, the composition.

**Figure 4.**Fixed point search for a black box with one and a black box with two fixed points. (

**a**) The output x is the fixed point c added to the input a. (

**b**) Circuit for finding a fixed point for a black box with two fixed points.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Baumeler, Ä.; Wolf, S. Non-Causal Computation. *Entropy* **2017**, *19*, 326.
https://doi.org/10.3390/e19070326

**AMA Style**

Baumeler Ä, Wolf S. Non-Causal Computation. *Entropy*. 2017; 19(7):326.
https://doi.org/10.3390/e19070326

**Chicago/Turabian Style**

Baumeler, Ämin, and Stefan Wolf. 2017. "Non-Causal Computation" *Entropy* 19, no. 7: 326.
https://doi.org/10.3390/e19070326