arXiv:1306.0510 · Physical Review Letters

Embedding Quantum Simulators for Quantum Computation of Entanglement

Roberto Di Candia, Bernabé Mejía, Hernán Castillo, Julen Simon Pedernales, Jorge Casanova, E. Solano

Published Dec 9, 2013• 26 citations

Abstract

We introduce the concept of embedding quantum simulators, a paradigm allowing the efficient quantum computation of a class of bipartite and multipartite entanglement monotones. It consists in the suitable encoding of a simulated quantum dynamics in the enlarged Hilbert space of an embedding quantum simulator. In this manner, entanglement monotones are conveniently mapped onto physical observables, overcoming the necessity of full tomography and reducing drastically the experimental requirements. Furthermore, this method is directly applicable to pure states and, assisted by classical algorithms, to the mixed-state case. Finally, we expect that the proposed embedding framework paves the way for a general theory of enhanced one-to-one quantum simulators.

DOI: 10.1103/physrevlett.111.240502

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arXiv:1306.0510 · Physical Review Letters

Embedding Quantum Simulators for Quantum Computation of Entanglement

Roberto Di Candia, Bernabé Mejía, Hernán Castillo, Julen Simon Pedernales, Jorge Casanova, E. Solano

Published Dec 9, 2013• 26 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevlett.111.240502

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We prove theoretical convergence results for these predictions in the large-qubit limit, $n\ensuremath{\rightarrow}\ensuremath{\infty}$. In light of these properties, QECD provides a consistent simulator of the evolution of classical observables, realized through projective quantum measurement, which is able to simulate spaces of classical observables of dimension ${2}^{n}$ using circuits of size $O({n}^{2})$ . We demonstrate the consistency of the scheme in prototypical dynamical systems involving periodic and quasiperiodic oscillators on tori. These examples include simulated quantum circuit experiments in Qiskit Aer, as well as actual experiments on the IBM Quantum System One.We develop a framework for simulating measure-preserving, ergodic dynamical systems on a quantum computer. Our approach provides an operator-theoretic representation of classical dynamics by combining ergodic theory with quantum information science. 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To achieve a quantum computational advantage, we project the state of the quantum system onto a finite-rank density operator on a ${2}^{n}$-dimensional tensor product Hilbert space associated with $n$ qubits. By employing discrete Fourier-Walsh transforms of spectral functions, the evolution operator of the finite-dimensional quantum system is factorized into tensor product form, enabling implementation through an $n$-channel quantum circuit of size $O(n)$ and no interchannel communication. Furthermore, the circuit features a state preparation stage, also of size $O(n)$, and a quantum Fourier transform stage of size $O({n}^{2})$, which makes predictions of observables possible by measurement in the standard computational basis. We prove theoretical convergence results for these predictions in the large-qubit limit, $n\ensuremath{\rightarrow}\ensuremath{\infty}$. 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Embedding quantum simulators

Roberto Di Candia

Quantum simulations consist in the reproduction of the dynamics of a quantum system on a controllable platform, with the goal of capturing an interesting feature of the considered model.It is broadly believed that the advent of quantum simulators will represent a technological revolution, as they promise to solve several problems which are considered intractable in a classical computer.Although there are strong theoretical bases confirming this claim, several aspects of quantum simulators have still to be studied, in order to faithfully prove their feasibility.Moreover, the general question on which features of the considered models are simulatable is an attractive research topic, whose study would help to define the limits of a quantum simulator.In this Thesis, we develop several algorithms, which are able to catch relevant properties of the simulated quantum model.The proposed protocols follow a new concept named embedding quantum simulator, in which the simulated Schrödinger equation is mapped onto an enlarged Hilbert space in a nontrivial way.Via this embedding, we are able to retrieve, by measuring few observables, quantities that generally require full tomography in order to be evaluated.Moreover, we pay a special attention to the experimental feasibility, defining mappings which are space e cient, and do not require the implementation of challenging Hamiltonians.The presented algorithms are general, and they may be implemented in several quantum platforms, e.g.photonics, trapped ions, circuit QED, among others.First, we propose a protocol which simulate the dynamics of an embedded Hamiltonian, allowing for the e cient extraction of a class of entanglement monotones.This is done using an embedding that is able to implement unphysical operations, as is the case of complex conjugation.The analysis is accompanied with a study of feasibility in a trapped-ion setup, which can be generalised to other platforms following similar computational models.Second, we propose an algorithm to measure n-time correlation functions of spinorial, fermionic, and bosonic operators, by considerably improving previous versions of the same result.We apply this protocol to the computation of magnetic susceptibilities, as well as to the simulation of Markovian and non-Markovian dissipative processes in a novel way, without the necessity of engineering any bath.All the proposed protocols are designed with a single ancillary qubit, minimising the needed experimental resources.We believe that embedding quantum simulators have a potential to become a powerful tool in the quantum simulation theory, since they pave the way for improving the flexibility of a quantum simulator in di↵erent experimental contexts.Resumen'' !!!En! esta!Tesis,!introducimos!el! concepto!de! "Embedding!Quantum!Simulator"! (EQS),!un!paradigma!que!nos!permite!capturar!características!especificas!de!un! modelo!cuántico,!cuya! medida!típicamente!supone!un! desafío!en! un! simulador!cuántico!estándar!("one@to@one!quantum!simulator"),!donde!la! dinámica!se! implementa!directamente.! El! "Embedding!Quantum!Simulator"! consiste!en! la! codificación!adecuada!de! la! dinámica!simulada!en! un! espacio!de! Hilbert! ampliado.! De! esta! manera,!cantidades!físicas!interesantes!están!convenientemente!mapeadas!a! observables!físicos,!superando!la! necesidad!de! tomografía!cuántica!y! ganando!en! términos!de! eficiencia.! !Los! protocolos! propuestos!son!muy!generales,!en!el!sentido!que!pueden!ser!implementados!en! plataformas!cuánticas!que! siguen!modelos!computacionales!típicos.! De! hecho,!una! característica!del! "Embedding!Quantum!Simulator"! es! que! puede!ser! aplicado!a! modelos!cuánticos!generales,!con! un! pequeño!coste!en! términos!de! recursos!experimentales!adicionales.! En! concreto,!hemos!diseñado!mapeos!capaces!de!capturar!la!dinámica!de!medidas!de!entrelazamiento!y!para!medir!las! funciones!de!correlación!temporales!en!sistemas!bosónicos!y!fermiónicos.!En!el! caso!de!las!medidas!de!entrelazamiento,!hemos!proporcionado!una!propuesta!de! implementación!realista!en!plataformas!basadas!en!iones!atrapados,!teniendo!en! cuenta!del! ruido!típico!de! estos!sistemas.! A! su! vez,! el! protocolo!para! calcular!funciones!de!correlación!ha!sido!aplicado!a!la!computación!de!susceptibilidades!magnéticas,!y! en! general!al! calculo!de! funciones!de! respuesta!lineales!y! no! lineales.! Por! otra! parte,!hemos!propuesto!un! nuevo!algoritmo!para! simular!sistemas!disipativos!sin! necesidad!de! hacer!ingeniería!de! baños.! Esto! nos! ha! permitido!introducir!un! nuevo!concepto!de! simulador!cuántico,!donde!no! queremos!crear!el! estado!final!bajo! una! dinámica!dada,!sino! que! apuntamos!directamente!al! valor!esperado!de! un! observable.! Por! tanto!el! EQS! es! potencialmente!útil! en! varios!campos,!como! la! materia!condensada,!química!cuántica,!óptica!cuántica,!etc.!! Esta!Tesis!contiene!un!Capitulo!introductorio,!seguido!de!cuatro!Capítulos.!Cada!uno! contiene!ejemplos!de! aplicaciones!del! "Embedding!Quantum!Simulator".! Concluimos! con! una! sección!de! Apéndices,! que! contiene!las! demostraciones!técnicas!de!las!afirmaciones!de!la!parte!principal.! En!los!Capítulos!2!y!3,!hemos!estudiado!un!protocolo!para!simular!la!dinámica!de! una! clase!de! medidas!de! entrelazamiento!en! sistemas!de! qubits.! El! mapeo!propuesto!puede!ser! implementado!en! una! plataforma!cuántica!añadiendo!un! solo! qubit,!y! la! longitud!de! interacción!de! !la! dinámica!se! incrementa!en! uno.! Todo!el!sistema!tiene!que!interactuar!con!el!qubit!auxiliar,!y!eso!puede!dar!lugar!a! interacciones! no! locales.! Hemos! mostrado!como! resolver!este! problema,!mediante!la!definición!de!un!qubit!lógico!a!costa!de!eficiencia!espacial,!es!decir!del! numero!de! partículas.! Por! último,!generalizamos! los! resultados!al! caso! de! matrices!densidad,!discutiendo!un! algoritmo!híbrido!clásico@cuántico.! Aunque!hemos!tratado!el! caso! de! las! medidas!de! entrelazamiento,!es! importante!mencionar!que! el! protocolo!propuesto!es! capaz!de! simular!operadores!antilineales!generales,!que! no! se! pueden!medir!en! un! "one@to@one!quantum!simulator".! En! el! Capitulo!3,! hemos!propuesto!una! de! estas!ideas!implementación!en! plataformas!basadas!en! iones!atrapados.! Este! Capitulo!proporciona!también!un! protocolo!para! medir!un! producto!tensorial!arbitrario!de! matrices!de! Pauli,!mediante!su! codificación!en! un! observable!de! un! qubit!! auxiliar.!El!análisis!vale!en!general!para!plataformas!cuánticas!donde!puertas!de! Mølmer@Sørensen!pueden!implementarse! de! manera!eficiente,!como! es! el! caso! de!la!óptica!lineal.!! Una! futura!investigación! interesante!de! estos!resultados!sería!el! estudio!de! las! propiedades!del! mapeo!propuesto,!con! el! fin! de! aumentar!la! flexibilidad!de! un! simulador!cuántico.! De! hecho,!si! no! estamos!limitados!a! un! solo! qubit!auxiliar,!mapeos!arbitrarios!a! espacio!de! Hilbert! más! grandes!podrían!simular!otras!cantidades!no!físicamente!accesible!en!un!"one@to@one!quantum!simulator".!Otra!cuestión!es! cómo! el! entorno!afecta!a! los! resultados!finales!de! la! dinámica!unitaria!simulada.! Aquí,! el! "Embedding!Quantum!Simulator"! podría!conducir!a! una!mejora!de!la!estabilidad!frente!al!ruido.!Estas!preguntas!se!quedan!abiertas,!y!requieren!de!mas!análisis!para!se!bien!entendida.!! Un!experimento!de!primeros!principios!basado!en!estas!ideas!se!está!ejecutando!en! este! momento!en! el! grupo!del! Prof.! Andrew!White! de! la! Universidad!de! Queensland! (Brisbane,!Australia).! La! implementación!es! en! una! plataforma!de! fotónica,!y!consiste!en!realizar!medidas!de!entrelazamiento!entre!dos!qubits!que! evolucionan!bajo! una! dinámica!específica.! El! experimento!se! realiza!con! tres! qubits,!cada! uno! de! ellos!correspondiente!a! una! polarización!de! la! señal!óptica!que!se!propaga.!Todas!las!operaciones!se!implementan!con!dispositivos!estándar!de!óptica!cuántica,!por!ejemplo!divisores!de!haz,!rotaciones!de!qubits!y!puertas!NOT! controladas.! Resultados! preliminares!muestran!que! la! medida!de! entrelazamiento!elegida!posee!una! alta! fidelidad,!y! eso! puede!llevar!a! implementaciones!similares!en!otras!plataformas!basadas!en!iones!atrapados!o! circuitos!superconductores.! En! el! Capitulo!4,! hemos!desarrollado!un! protocolo!para! computar!funciones!de! correlación!temporal!de! operadores!generales!en! un! simulador!cuántico.! También! en! este! caso,!hemos!mapeado!a! una! ecuación!de! Schrödinger!en! un! espacio!de!Hilbert!de!dimensión!doble.!El!mapeo!tiene!la!misma!estructura!que! el!caso!de!los!operadores!antilineales,!discutido!en!los!Capitulos!2!y!3,!y!esto!es! un!indicio!que!otras!aplicaciones!no!triviales!son!posibles.!Hemos!discutido!cómo!aplicar!el! protocolo!en! los! casos!espinorial,!fermionico!y! bosónicos,!mostrando!que! el! método!es! eficiente!en! términos!de! tiempo!y! de! espacio.! El! algoritmo!propuesto!no! requiere!la! implementación!de! Hamiltonianos! controlados,!que! puede!ser! un! problema!complicado!para! la! mayoría!de! modelos!interesante!desde!el! punto!de! vista!físico.! Este! aspecto,!en! comparación!con! los! protocolos!anteriores,!conlleva!una!ganancia!enorme!a!nivel!experimental,!y!es!posible!que! pronto!veamos!experimentos!de! primeros!principios!donde!se! aplica!este! protocolo.!También!en!este!caso!necesitamos!que!el!sistema!interactúe!a!nivel!no! local!con!un!qubit!auxiliar.!Este!problema!se!puede!resolver!de!la!misma!manera!que!en!el!Capitulo!1,!codificando!el!qubit!auxiliar!en!una!serie!de!qubits!lógicos.! Como! aplicación! típica,!hemos!considerado!la! simulación!cuántica!de! susceptibilidades!magnéticas!y!de!funciones!de!respuesta!lineales!y!no@lineales.! Este! protocolo!es! suficientemente!simple!para! permitir!una! implementación!experimental!con!la!tecnología!actual!o!en!un!futuro!próximo,!dependiendo!de!la! plataforma.! En! el! Capitulo!5,! hemos!estudiado!un! protocolo!original!para! simular!procesos!disipativos!MarkQuantum simulations consist in the reproduction of the dynamics of a quantum system on a controllable platform, with the goal of capturing an interesting feature of the considered model.It is broadly believed that the advent of quantum simulators will represent a technological revolution, as they promise to solve several problems which are considered intractable in a classical computer.Although there are strong theoretical bases confirming this claim, several aspects of quantum simulators have still to be studied, in order to faithfully prove their feasibility.Moreover, the general question on which features of the considered models are simulatable is an attractive research topic, whose study would help to define the limits of a quantum simulator.In this Thesis, we develop several algorithms, which are able to catch relevant properties of the simulated quantum model.The proposed protocols follow a new concept named embedding quantum simulator, in which the simulated Schrödinger equation is mapped onto an enlarged Hilbert space in a nontrivial way.Via this embedding, we are able to retrieve, by measuring few observables, quantities that generally require full tomography in order to be evaluated.Moreover, we pay a special attention to the experimental feasibility, defining mappings which are space e cient, and do not require the implementation of challenging Hamiltonians.The presented algorithms are general, and they may be implemented in several quantum platforms, e.g.photonics, trapped ions, circuit QED, among others.First, we propose a protocol which simulate the dynamics of an embedded Hamiltonian, allowing for the e cient extraction of a class of entanglement monotones.This is done using an embedding that is able to implement unphysical operations, as is the case of complex conjugation.The analysis is accompanied with a study of feasibility in a trapped-ion setup, which can be generalised to other platforms following similar computational models.Second, we propose an algorithm to measure n-time correlation functions of spinorial, fermionic, and bosonic operators, by considerably improving previous versions of the same result.We apply this protocol to the computation of magnetic susceptibilities, as well as to the simulation of Markovian and non-Markovian dissipative processes in a novel way, without the necessity of engineering any bath.All the proposed protocols are designed with a single ancillary qubit, minimising the needed experimental resources.We believe that embedding quantum simulators have a potential to become a powerful tool in the quantum simulation theory, since they pave the way for improving the flexibility of a quantum simulator in di↵erent experimental contexts.Resumen'' !!!En! esta!Tesis,!introducimos!el! concepto!de! "Embedding!Quantum!Simulator"! (EQS),!un!paradigma!que!nos!permite!capturar!características!especificas!de!un! modelo!cuántico,!cuya! medida!típicamente!supone!un! desafío!en! un! simulador!cuántico!estándar!("one@to@one!quantum!simulator"),!donde!la! dinámica!se! implementa!directamente.! El! "Embedding!Quantum!Simulator"! consiste!en! la! codificación!adecuada!de! la! dinámica!simulada!en! un! espacio!de! Hilbert! ampliado.! De! esta! manera,!cantidades!físicas!interesantes!están!convenientemente!mapeadas!a! observables!físicos,!superando!la! necesidad!de! tomografía!cuántica!y! ganando!en! términos!de! eficiencia.! !Los! protocolos! propuestos!son!muy!generales,!en!el!sentido!que!pueden!ser!implementados!en! plataformas!cuánticas!que! siguen!modelos!computacionales!típicos.! De! hecho,!una! característica!del! "Embedding!Quantum!Simulator"! es! que! puede!ser! aplicado!a! modelos!cuánticos!generales,!con! un! pequeño!coste!en! términos!de! recursos!experimentales!adicionales.! En! concreto,!hemos!diseñado!mapeos!capaces!de!capturar!la!dinámica!de!medidas!de!entrelazamiento!y!para!medir!las! funciones!de!correlación!temporales!en!sistemas!bosónicos!y!fermiónicos.!En!el! caso!de!las!medidas!de!entrelazamiento,!hemos!proporcionado!una!propuesta!de! implementación!realista!en!plataformas!basadas!en!iones!atrapados,!teniendo!en! cuenta!del! ruido!típico!de! estos!sistemas.! A! su! vez,! el! protocolo!para! calcular!funciones!de!correlación!ha!sido!aplicado!a!la!computación!de!susceptibilidades!magnéticas,!y! en! general!al! calculo!de! funciones!de! respuesta!lineales!y! no! lineales.! Por! otra! parte,!hemos!propuesto!un! nuevo!algoritmo!para! simular!sistemas!disipativos!sin! necesidad!de! hacer!ingeniería!de! baños.! Esto! nos! ha! permitido!introducir!un! nuevo!concepto!de! simulador!cuántico,!donde!no! queremos!crear!el! estado!final!bajo! una! dinámica!dada,!sino! que! apuntamos!directamente!al! valor!esperado!de! un! observable.! Por! tanto!el! EQS! es! potencialmente!útil! en! varios!campos,!como! la! materia!condensada,!química!cuántica,!óptica!cuántica,!etc.!! Esta!Tesis!contiene!un!Capitulo!introductorio,!seguido!de!cuatro!Capítulos.!Cada!uno! contiene!ejemplos!de! aplicaciones!del! "Embedding!Quantum!Simulator".! Concluimos! con! una! sección!de! Apéndices,! que! contiene!las! demostraciones!técnicas!de!las!afirmaciones!de!la!parte!principal.! En!los!Capítulos!2!y!3,!hemos!estudiado!un!protocolo!para!simular!la!dinámica!de! una! clase!de! medidas!de! entrelazamiento!en! sistemas!de! qubits.! El! mapeo!propuesto!puede!ser! implementado!en! una! plataforma!cuántica!añadiendo!un! solo! qubit,!y! la! longitud!de! interacción!de! !la! dinámica!se! incrementa!en! uno.! Todo!el!sistema!tiene!que!interactuar!con!el!qubit!auxiliar,!y!eso!puede!dar!lugar!a! interacciones! no! locales.! Hemos! mostrado!como! resolver!este! problema,!mediante!la!definición!de!un!qubit!lógico!a!costa!de!eficiencia!espacial,!es!decir!del! numero!de! partículas.! Por! último,!generalizamos! los! resultados!al! caso! de! matrices!densidad,!discutiendo!un! algoritmo!híbrido!clásico@cuántico.! Aunque!hemos!tratado!el! caso! de! las! medidas!de! entrelazamiento,!es! importante!mencionar!que! el! protocolo!propuesto!es! capaz!de! simular!operadores!antilineales!generales,!que! no! se! pueden!medir!en! un! "one@to@one!quantum!simulator".! En! el! Capitulo!3,! hemos!propuesto!una! de! estas!ideas!implementación!en! plataformas!basadas!en! iones!atrapados.! Este! Capitulo!proporciona!también!un! protocolo!para! medir!un! producto!tensorial!arbitrario!de! matrices!de! Pauli,!mediante!su! codificación!en! un! observable!de! un! qubit!! auxiliar.!El!análisis!vale!en!general!para!plataformas!cuánticas!donde!puertas!de! Mølmer@Sørensen!pueden!implementarse! de! manera!eficiente,!como! es! el! caso! de!la!óptica!lineal.!! Una! futura!investigación! interesante!de! estos!resultados!sería!el! estudio!de! las! propiedades!del! mapeo!propuesto,!con! el! fin! de! aumentar!la! flexibilidad!de! un! simulador!cuántico.! De! hecho,!si! no! estamos!limitados!a! un! solo! qubit!auxiliar,!mapeos!arbitrarios!a! espacio!de! Hilbert! más! grandes!podrían!simular!otras!cantidades!no!físicamente!accesible!en!un!"one@to@one!quantum!simulator".!Otra!cuestión!es! cómo! el! entorno!afecta!a! los! resultados!finales!de! la! dinámica!unitaria!simulada.! Aquí,! el! "Embedding!Quantum!Simulator"! podría!conducir!a! una!mejora!de!la!estabilidad!frente!al!ruido.!Estas!preguntas!se!quedan!abiertas,!y!requieren!de!mas!análisis!para!se!bien!entendida.!! Un!experimento!de!primeros!principios!basado!en!estas!ideas!se!está!ejecutando!en! este! momento!en! el! grupo!del! Prof.! Andrew!White! de! la! Universidad!de! Queensland! (Brisbane,!Australia).! La! implementación!es! en! una! plataforma!de! fotónica,!y!consiste!en!realizar!medidas!de!entrelazamiento!entre!dos!qubits!que! evolucionan!bajo! una! dinámica!específica.! El! experimento!se! realiza!con! tres! qubits,!cada! uno! de! ellos!correspondiente!a! una! polarización!de! la! señal!óptica!que!se!propaga.!Todas!las!operaciones!se!implementan!con!dispositivos!estándar!de!óptica!cuántica,!por!ejemplo!divisores!de!haz,!rotaciones!de!qubits!y!puertas!NOT! controladas.! Resultados! preliminares!muestran!que! la! medida!de! entrelazamiento!elegida!posee!una! alta! fidelidad,!y! eso! puede!llevar!a! implementaciones!similares!en!otras!plataformas!basadas!en!iones!atrapados!o! circuitos!superconductores.! En! el! Capitulo!4,! hemos!desarrollado!un! protocolo!para! computar!funciones!de! correlación!temporal!de! operadores!generales!en! un! simulador!cuántico.! También! en! este! caso,!hemos!mapeado!a! una! ecuación!de! Schrödinger!en! un! espacio!de!Hilbert!de!dimensión!doble.!El!mapeo!tiene!la!misma!estructura!que! el!caso!de!los!operadores!antilineales,!discutido!en!los!Capitulos!2!y!3,!y!esto!es! un!indicio!que!otras!aplicaciones!no!triviales!son!posibles.!Hemos!discutido!cómo!aplicar!el! protocolo!en! los! casos!espinorial,!fermionico!y! bosónicos,!mostrando!que! el! método!es! eficiente!en! términos!de! tiempo!y! de! espacio.! El! algoritmo!propuesto!no! requiere!la! implementación!de! Hamiltonianos! controlados,!que! puede!ser! un! problema!complicado!para! la! mayoría!de! modelos!interesante!desde!el! punto!de! vista!físico.! Este! aspecto,!en! comparación!con! los! protocolos!anteriores,!conlleva!una!ganancia!enorme!a!nivel!experimental,!y!es!posible!que! pronto!veamos!experimentos!de! primeros!principios!donde!se! aplica!este! protocolo.!También!en!este!caso!necesitamos!que!el!sistema!interactúe!a!nivel!no! local!con!un!qubit!auxiliar.!Este!problema!se!puede!resolver!de!la!misma!manera!que!en!el!Capitulo!1,!codificando!el!qubit!auxiliar!en!una!serie!de!qubits!lógicos.! Como! aplicación! típica,!hemos!considerado!la! simulación!cuántica!de! susceptibilidades!magnéticas!y!de!funciones!de!respuesta!lineales!y!no@lineales.! Este! protocolo!es! suficientemente!simple!para! permitir!una! implementación!experimental!con!la!tecnología!actual!o!en!un!futuro!próximo,!dependiendo!de!la! plataforma.! En! el! Capitulo!5,! hemos!estudiado!un! protocolo!original!para! simular!procesos!disipativos!Mark

Bulletin of the American Physical SocietyMar 6, 20140 citations

We propose a quantum algorithm in an embedding ion-trap quantum simulator for the efficient computation of

N\text{-qubit}

entanglement monotones without the necessity of full tomography. Moreover, we discuss possible realistic scenarios and study the associated decoherence mechanisms.

\mathcal{H}

. Furthermore, an embedding of classical observables into self-adjoint operators on

\mathcal{H}

is established, such that quantum mechanical expectation values are consistent with pointwise function evaluation. In this scheme, quantum states and observables evolve unitarily under the lifted action of Koopman evolution operators of the classical system. Moreover, by virtue of the reproducing property of

\mathcal{H}

, the quantum system is pointwise-consistent with the underlying classical dynamics. To achieve a quantum computational advantage, we project the state of the quantum system onto a finite-rank density operator on a

{2}^{n}

-dimensional tensor product Hilbert space associated with

n

qubits. By employing discrete Fourier-Walsh transforms of spectral functions, the evolution operator of the finite-dimensional quantum system is factorized into tensor product form, enabling implementation through an

n

-channel quantum circuit of size

O(n)

and no interchannel communication. Furthermore, the circuit features a state preparation stage, also of size

O(n)

, and a quantum Fourier transform stage of size

O({n}^{2})

, which makes predictions of observables possible by measurement in the standard computational basis. We prove theoretical convergence results for these predictions in the large-qubit limit, $n\ensuremath{\rightarrow}\ensuremath{\infty}$. In light of these properties, QECD provides a consistent simulator of the evolution of classical observables, realized through projective quantum measurement, which is able to simulate spaces of classical observables of dimension

{2}^{n}

using circuits of size

O({n}^{2})

. We demonstrate the consistency of the scheme in prototypical dynamical systems involving periodic and quasiperiodic oscillators on tori. These examples include simulated quantum circuit experiments in Qiskit Aer, as well as actual experiments on the IBM Quantum System One.