Quantum circuits for discrete-time quantum walks with position-dependent coin operator
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2022
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866913708537544704 |
|---|---|
| author | Nzongani, Ugo Zylberman, Julien Doncecchi, Carlo-Elia Pérez, Armando Debbasch, Fabrice Arnault, Pablo |
| author_facet | Nzongani, Ugo Zylberman, Julien Doncecchi, Carlo-Elia Pérez, Armando Debbasch, Fabrice Arnault, Pablo |
| contents | The aim of this paper is to build quantum circuits that implement discrete-time quantum walks having an arbitrary position-dependent coin operator. The position of the walker is encoded in base 2: with $n$ wires, each corresponding to one qubit, we encode $2^n$ position states. The data necessary to define an arbitrary position-dependent coin operator is therefore exponential in $n$.
We first propose a circuit implementing the position-dependent coin operator, that is naive, in the sense that it has exponential depth and implements sequentially all appropriate position-dependent coin operators. We then propose a circuit that "transfers" all the depth into ancillae, yielding a final depth that is linear in $n$ at the cost of an exponential number of ancillae. The main idea of this linear-depth circuit is to implement in parallel all coin operators at the different positions.
Finally, we extend the result of Ref. [2] from position-dependent unitaries which are diagonal in the position basis to position-dependent $2 \times 2$-block-diagonal unitaries: indeed, we show that for a position dependence of the coin operator (the block-diagonal unitary) which is smooth enough, one can find an efficient quantum-circuit implementation approximating the coin operator up to an error $ε$ (in terms of the spectral norm), the depth and size of which scale as $O(1/ε)$.
A typical application of the efficient implementation would be the quantum simulation of a relativistic spin-1/2 particle on a lattice, coupled to a smooth external gauge field; notice that recently, quantum spatial-search schemes have been developed which use gauge fields as the oracle, to mark the vertex to be found [3, 4]. A typical application of the linear-depth circuit would be when there is spatial noise on the coin operator (and hence a non-smooth dependence in the position). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_05271 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Quantum circuits for discrete-time quantum walks with position-dependent coin operator Nzongani, Ugo Zylberman, Julien Doncecchi, Carlo-Elia Pérez, Armando Debbasch, Fabrice Arnault, Pablo Quantum Physics Emerging Technologies The aim of this paper is to build quantum circuits that implement discrete-time quantum walks having an arbitrary position-dependent coin operator. The position of the walker is encoded in base 2: with $n$ wires, each corresponding to one qubit, we encode $2^n$ position states. The data necessary to define an arbitrary position-dependent coin operator is therefore exponential in $n$. We first propose a circuit implementing the position-dependent coin operator, that is naive, in the sense that it has exponential depth and implements sequentially all appropriate position-dependent coin operators. We then propose a circuit that "transfers" all the depth into ancillae, yielding a final depth that is linear in $n$ at the cost of an exponential number of ancillae. The main idea of this linear-depth circuit is to implement in parallel all coin operators at the different positions. Finally, we extend the result of Ref. [2] from position-dependent unitaries which are diagonal in the position basis to position-dependent $2 \times 2$-block-diagonal unitaries: indeed, we show that for a position dependence of the coin operator (the block-diagonal unitary) which is smooth enough, one can find an efficient quantum-circuit implementation approximating the coin operator up to an error $ε$ (in terms of the spectral norm), the depth and size of which scale as $O(1/ε)$. A typical application of the efficient implementation would be the quantum simulation of a relativistic spin-1/2 particle on a lattice, coupled to a smooth external gauge field; notice that recently, quantum spatial-search schemes have been developed which use gauge fields as the oracle, to mark the vertex to be found [3, 4]. A typical application of the linear-depth circuit would be when there is spatial noise on the coin operator (and hence a non-smooth dependence in the position). |
| title | Quantum circuits for discrete-time quantum walks with position-dependent coin operator |
| topic | Quantum Physics Emerging Technologies |
| url | https://arxiv.org/abs/2211.05271 |