Pub Date : 2024-07-10DOI: 10.22331/q-2024-07-10-1400
Campbell McLauchlan, Benjamin Béri
Majorana zero modes (MZMs) are promising candidates for topologically-protected quantum computing hardware, however their large-scale use will likely require quantum error correction. Majorana surface codes (MSCs) have been proposed to achieve this. However, many MSC properties remain unexplored. We present a unified framework for MSC "twist defects" $unicode{x2013}$ anyon-like objects encoding quantum information. We show that twist defects in MSCs can encode twice the amount of topologically protected information as in qubit-based codes or other MSC encoding schemes. This is due to twists encoding both logical qubits and "logical MZMs," with the latter enhancing the protection microscopic MZMs can offer. We explain how to perform universal computation with logical qubits and logical MZMs while potentially using far fewer resources than in other MSC schemes. All Clifford gates can be implemented on logical qubits by braiding twist defects. We introduce lattice-surgery-based techniques for computing with logical MZMs and logical qubits, achieving the effect of Clifford gates with zero time overhead. We also show that logical MZMs may result in improved spatial overheads for sufficiently low rates of quasi-particle poisoning. Finally, we introduce a novel MSC analogue of transversal gates that achieves encoded Clifford gates in small codes by braiding microscopic MZMs. MSC twist defects thus open new paths towards fault-tolerant quantum computation.
{"title":"A new twist on the Majorana surface code: Bosonic and fermionic defects for fault-tolerant quantum computation","authors":"Campbell McLauchlan, Benjamin Béri","doi":"10.22331/q-2024-07-10-1400","DOIUrl":"https://doi.org/10.22331/q-2024-07-10-1400","url":null,"abstract":"Majorana zero modes (MZMs) are promising candidates for topologically-protected quantum computing hardware, however their large-scale use will likely require quantum error correction. Majorana surface codes (MSCs) have been proposed to achieve this. However, many MSC properties remain unexplored. We present a unified framework for MSC \"twist defects\" $unicode{x2013}$ anyon-like objects encoding quantum information. We show that twist defects in MSCs can encode twice the amount of topologically protected information as in qubit-based codes or other MSC encoding schemes. This is due to twists encoding both logical qubits and \"logical MZMs,\" with the latter enhancing the protection microscopic MZMs can offer. We explain how to perform universal computation with logical qubits and logical MZMs while potentially using far fewer resources than in other MSC schemes. All Clifford gates can be implemented on logical qubits by braiding twist defects. We introduce lattice-surgery-based techniques for computing with logical MZMs and logical qubits, achieving the effect of Clifford gates with zero time overhead. We also show that logical MZMs may result in improved spatial overheads for sufficiently low rates of quasi-particle poisoning. Finally, we introduce a novel MSC analogue of transversal gates that achieves encoded Clifford gates in small codes by braiding microscopic MZMs. MSC twist defects thus open new paths towards fault-tolerant quantum computation.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":null,"pages":null},"PeriodicalIF":6.4,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141566268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-10DOI: 10.22331/q-2024-07-10-1404
Fereshte Shahbeigi, Christopher T. Chubb, Ryszard Kukulski, Łukasz Pawela, Kamil Korzekwa
The classical embeddability problem asks whether a given stochastic matrix $T$, describing transition probabilities of a $d$-level system, can arise from the underlying homogeneous continuous-time Markov process. Here, we investigate the quantum version of this problem, asking of the existence of a Markovian quantum channel generating state transitions described by a given $T$. More precisely, we aim at characterising the set of quantum-embeddable stochastic matrices that arise from memoryless continuous-time quantum evolution. To this end, we derive both upper and lower bounds on that set, providing new families of stochastic matrices that are quantum-embeddable but not classically-embeddable, as well as families of stochastic matrices that are not quantum-embeddable. As a result, we demonstrate that a larger set of transition matrices can be explained by memoryless models if the dynamics is allowed to be quantum, but we also identify a non-zero measure set of random processes that cannot be explained by either classical or quantum memoryless dynamics. Finally, we fully characterise extreme stochastic matrices (with entries given only by zeros and ones) that are quantum-embeddable.
{"title":"Quantum-embeddable stochastic matrices","authors":"Fereshte Shahbeigi, Christopher T. Chubb, Ryszard Kukulski, Łukasz Pawela, Kamil Korzekwa","doi":"10.22331/q-2024-07-10-1404","DOIUrl":"https://doi.org/10.22331/q-2024-07-10-1404","url":null,"abstract":"The classical embeddability problem asks whether a given stochastic matrix $T$, describing transition probabilities of a $d$-level system, can arise from the underlying homogeneous continuous-time Markov process. Here, we investigate the quantum version of this problem, asking of the existence of a Markovian quantum channel generating state transitions described by a given $T$. More precisely, we aim at characterising the set of quantum-embeddable stochastic matrices that arise from memoryless continuous-time quantum evolution. To this end, we derive both upper and lower bounds on that set, providing new families of stochastic matrices that are quantum-embeddable but not classically-embeddable, as well as families of stochastic matrices that are not quantum-embeddable. As a result, we demonstrate that a larger set of transition matrices can be explained by memoryless models if the dynamics is allowed to be quantum, but we also identify a non-zero measure set of random processes that cannot be explained by either classical or quantum memoryless dynamics. Finally, we fully characterise extreme stochastic matrices (with entries given only by zeros and ones) that are quantum-embeddable.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":null,"pages":null},"PeriodicalIF":6.4,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141584495","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-10DOI: 10.22331/q-2024-07-10-1403
Michael Liaofan Liu, Nathanan Tantivasadakarn, Victor V. Albert
The CSS code construction is a powerful framework used to express features of a quantum code in terms of a pair of underlying classical codes. Its subsystem extension allows for similar expressions, but the general case has not been fully explored. Extending previous work of Aly, Klappenecker, and Sarvepalli [5], we determine subsystem CSS code parameters, express codewords, and develop a Steane-type decoder using only data from the two underlying classical codes. Generalizing a result of Kovalev and Pryadko [55], we show that any subsystem stabilizer code can be "doubled" to yield a subsystem CSS code with twice the number of physical, logical, and gauge qudits and up to twice the code distance. This mapping preserves locality and is tighter than the Majorana-based mapping of Bravyi, Terhal, and Leemhuis [19]. Using Goursat's Lemma, we show that every subsystem stabilizer code can be constructed from two nested subsystem CSS codes satisfying certain constraints, and we characterize subsystem stabilizer codes based on the nested codes' properties.
{"title":"Subsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat’s Lemma","authors":"Michael Liaofan Liu, Nathanan Tantivasadakarn, Victor V. Albert","doi":"10.22331/q-2024-07-10-1403","DOIUrl":"https://doi.org/10.22331/q-2024-07-10-1403","url":null,"abstract":"The CSS code construction is a powerful framework used to express features of a quantum code in terms of a pair of underlying classical codes. Its subsystem extension allows for similar expressions, but the general case has not been fully explored. Extending previous work of Aly, Klappenecker, and Sarvepalli [5], we determine subsystem CSS code parameters, express codewords, and develop a Steane-type decoder using only data from the two underlying classical codes. Generalizing a result of Kovalev and Pryadko [55], we show that any subsystem stabilizer code can be \"doubled\" to yield a subsystem CSS code with twice the number of physical, logical, and gauge qudits and up to twice the code distance. This mapping preserves locality and is tighter than the Majorana-based mapping of Bravyi, Terhal, and Leemhuis [19]. Using Goursat's Lemma, we show that every subsystem stabilizer code can be constructed from two nested subsystem CSS codes satisfying certain constraints, and we characterize subsystem stabilizer codes based on the nested codes' properties.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":null,"pages":null},"PeriodicalIF":6.4,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141566272","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-04DOI: 10.22331/q-2024-07-04-1397
Gabriel Wong, Robert Raussendorf, Bartlomiej Czech
Measurement-Based Quantum Computation (MBQC) is a model of quantum computation, which uses local measurements instead of unitary gates. Here we explain that the MBQC procedure has a fundamental basis in an underlying gauge theory. This perspective provides a theoretical foundation for global aspects of MBQC. The gauge transformations reflect the freedom of formulating the same MBQC computation in different local reference frames. The main identifications between MBQC and gauge theory concepts are: (i) the computational output of MBQC is a holonomy of the gauge field, (ii) the adaptation of measurement basis that remedies the inherent randomness of quantum measurements is effected by gauge transformations. The gauge theory of MBQC also plays a role in characterizing the entanglement structure of symmetry-protected topologically (SPT) ordered states, which are resources for MBQC. Our framework situates MBQC in a broader context of condensed matter and high energy theory.
{"title":"The Gauge Theory of Measurement-Based Quantum Computation","authors":"Gabriel Wong, Robert Raussendorf, Bartlomiej Czech","doi":"10.22331/q-2024-07-04-1397","DOIUrl":"https://doi.org/10.22331/q-2024-07-04-1397","url":null,"abstract":"Measurement-Based Quantum Computation (MBQC) is a model of quantum computation, which uses local measurements instead of unitary gates. Here we explain that the MBQC procedure has a fundamental basis in an underlying gauge theory. This perspective provides a theoretical foundation for global aspects of MBQC. The gauge transformations reflect the freedom of formulating the same MBQC computation in different local reference frames. The main identifications between MBQC and gauge theory concepts are: (i) the computational output of MBQC is a holonomy of the gauge field, (ii) the adaptation of measurement basis that remedies the inherent randomness of quantum measurements is effected by gauge transformations. The gauge theory of MBQC also plays a role in characterizing the entanglement structure of symmetry-protected topologically (SPT) ordered states, which are resources for MBQC. Our framework situates MBQC in a broader context of condensed matter and high energy theory.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":null,"pages":null},"PeriodicalIF":6.4,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141521459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-04DOI: 10.22331/q-2024-07-04-1398
Jonathan Conrad, Jens Eisert, Jean-Pierre Seifert
We introduce a new class of random Gottesman-Kitaev-Preskill (GKP) codes derived from the cryptanalysis of the so-called NTRU cryptosystem. The derived codes are $good$ in that they exhibit constant rate and average distance scaling $Delta propto sqrt{n}$ with high probability, where $n$ is the number of bosonic modes, which is a distance scaling equivalent to that of a GKP code obtained by concatenating single mode GKP codes into a qubit-quantum error correcting code with linear distance. The derived class of NTRU-GKP codes has the additional property that $decoding$ for a stochastic displacement noise model is equivalent to $decrypting$ the NTRU cryptosystem, such that every random instance of the code naturally comes with an efficient decoder. This construction highlights how the GKP code bridges aspects of classical error correction, quantum error correction as well as post-quantum cryptography. We underscore this connection by discussing the computational hardness of decoding GKP codes and propose, as a new application, a simple public key quantum communication protocol with security inherited from the NTRU cryptosystem.
{"title":"Good Gottesman-Kitaev-Preskill codes from the NTRU cryptosystem","authors":"Jonathan Conrad, Jens Eisert, Jean-Pierre Seifert","doi":"10.22331/q-2024-07-04-1398","DOIUrl":"https://doi.org/10.22331/q-2024-07-04-1398","url":null,"abstract":"We introduce a new class of random Gottesman-Kitaev-Preskill (GKP) codes derived from the cryptanalysis of the so-called NTRU cryptosystem. The derived codes are $good$ in that they exhibit constant rate and average distance scaling $Delta propto sqrt{n}$ with high probability, where $n$ is the number of bosonic modes, which is a distance scaling equivalent to that of a GKP code obtained by concatenating single mode GKP codes into a qubit-quantum error correcting code with linear distance. The derived class of NTRU-GKP codes has the additional property that $decoding$ for a stochastic displacement noise model is equivalent to $decrypting$ the NTRU cryptosystem, such that every random instance of the code naturally comes with an efficient decoder. This construction highlights how the GKP code bridges aspects of classical error correction, quantum error correction as well as post-quantum cryptography. We underscore this connection by discussing the computational hardness of decoding GKP codes and propose, as a new application, a simple public key quantum communication protocol with security inherited from the NTRU cryptosystem.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":null,"pages":null},"PeriodicalIF":6.4,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141521460","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-04DOI: 10.22331/q-2024-07-04-1399
Yusuf Alnawakhtha, Atul Mantri, Carl A. Miller, Daochen Wang
Trapdoor claw-free functions (TCFs) are immensely valuable in cryptographic interactions between a classical client and a quantum server. Typically, a protocol has the quantum server prepare a superposition of two-bit strings of a claw and then measure it using Pauli-$X$ or $Z$ measurements. In this paper, we demonstrate a new technique that uses the entire range of qubit measurements from the $XY$-plane. We show the advantage of this approach in two applications. First, building on (Brakerski et al. 2018, Kalai et al. 2022), we show an optimized two-round proof of quantumness whose security can be expressed directly in terms of the hardness of the LWE (learning with errors) problem. Second, we construct a one-round protocol for blind remote preparation of an arbitrary state on the $XY$-plane up to a Pauli-$Z$ correction.
{"title":"Lattice-Based Quantum Advantage from Rotated Measurements","authors":"Yusuf Alnawakhtha, Atul Mantri, Carl A. Miller, Daochen Wang","doi":"10.22331/q-2024-07-04-1399","DOIUrl":"https://doi.org/10.22331/q-2024-07-04-1399","url":null,"abstract":"Trapdoor claw-free functions (TCFs) are immensely valuable in cryptographic interactions between a classical client and a quantum server. Typically, a protocol has the quantum server prepare a superposition of two-bit strings of a claw and then measure it using Pauli-$X$ or $Z$ measurements. In this paper, we demonstrate a new technique that uses the entire range of qubit measurements from the $XY$-plane. We show the advantage of this approach in two applications. First, building on (Brakerski et al. 2018, Kalai et al. 2022), we show an optimized two-round proof of quantumness whose security can be expressed directly in terms of the hardness of the LWE (learning with errors) problem. Second, we construct a one-round protocol for blind remote preparation of an arbitrary state on the $XY$-plane up to a Pauli-$Z$ correction.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":null,"pages":null},"PeriodicalIF":6.4,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141521461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-03DOI: 10.22331/q-2024-07-03-1395
Lorenzo Leone, Salvatore F.E. Oliviero, Lukasz Cincio, M. Cerezo
Variational Quantum Algorithms (VQAs) and Quantum Machine Learning (QML) models train a parametrized quantum circuit to solve a given learning task. The success of these algorithms greatly hinges on appropriately choosing an ansatz for the quantum circuit. Perhaps one of the most famous ansatzes is the one-dimensional layered Hardware Efficient Ansatz (HEA), which seeks to minimize the effect of hardware noise by using native gates and connectives. The use of this HEA has generated a certain ambivalence arising from the fact that while it suffers from barren plateaus at long depths, it can also avoid them at shallow ones. In this work, we attempt to determine whether one should, or should not, use a HEA. We rigorously identify scenarios where shallow HEAs should likely be avoided (e.g., VQA or QML tasks with data satisfying a volume law of entanglement). More importantly, we identify a Goldilocks scenario where shallow HEAs could achieve a quantum speedup: QML tasks with data satisfying an area law of entanglement. We provide examples for such scenario (such as Gaussian diagonal ensemble random Hamiltonian discrimination), and we show that in these cases a shallow HEA is always trainable and that there exists an anti-concentration of loss function values. Our work highlights the crucial role that input states play in the trainability of a parametrized quantum circuit, a phenomenon that is verified in our numerics.
变分量子算法（VQAs）和量子机器学习（QML）模型训练参数化的量子电路来解决给定的学习任务。这些算法的成功与否在很大程度上取决于是否为量子电路选择了合适的算式。一维分层硬件高效解析（HEA）也许是最著名的解析之一，它试图通过使用原生门和连接件将硬件噪声的影响降到最低。这种 HEA 的使用产生了一定的矛盾性，因为它在长深度时会出现贫瘠高原，但在浅深度时也能避免贫瘠高原。在这项工作中，我们试图确定是否应该使用 HEA。我们严格确定了应避免使用浅层 HEA 的情况（例如，数据满足纠缠体积定律的 VQA 或 QML 任务）。更重要的是，我们确定了浅层 HEA 可以实现量子加速的黄金组合方案：数据满足纠缠面积律的 QML 任务。我们举例说明了这种情况（如高斯对角集合随机哈密顿辨别），并证明在这些情况下，浅层 HEA 始终是可训练的，而且存在损失函数值的反集中。我们的工作强调了输入状态在参数化量子电路可训练性中的关键作用，这一现象在我们的数值计算中得到了验证。
{"title":"On the practical usefulness of the Hardware Efficient Ansatz","authors":"Lorenzo Leone, Salvatore F.E. Oliviero, Lukasz Cincio, M. Cerezo","doi":"10.22331/q-2024-07-03-1395","DOIUrl":"https://doi.org/10.22331/q-2024-07-03-1395","url":null,"abstract":"Variational Quantum Algorithms (VQAs) and Quantum Machine Learning (QML) models train a parametrized quantum circuit to solve a given learning task. The success of these algorithms greatly hinges on appropriately choosing an ansatz for the quantum circuit. Perhaps one of the most famous ansatzes is the one-dimensional layered Hardware Efficient Ansatz (HEA), which seeks to minimize the effect of hardware noise by using native gates and connectives. The use of this HEA has generated a certain ambivalence arising from the fact that while it suffers from barren plateaus at long depths, it can also avoid them at shallow ones. In this work, we attempt to determine whether one should, or should not, use a HEA. We rigorously identify scenarios where shallow HEAs should likely be avoided (e.g., VQA or QML tasks with data satisfying a volume law of entanglement). More importantly, we identify a Goldilocks scenario where shallow HEAs could achieve a quantum speedup: QML tasks with data satisfying an area law of entanglement. We provide examples for such scenario (such as Gaussian diagonal ensemble random Hamiltonian discrimination), and we show that in these cases a shallow HEA is always trainable and that there exists an anti-concentration of loss function values. Our work highlights the crucial role that input states play in the trainability of a parametrized quantum circuit, a phenomenon that is verified in our numerics.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":null,"pages":null},"PeriodicalIF":6.4,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141495860","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-03DOI: 10.22331/q-2024-07-03-1396
L. Sunil Chandran, Rishikesh Gajjala
The most efficient automated way to construct a large class of quantum photonic experiments is via abstract representation of graphs with certain properties. While new directions were explored using Artificial intelligence and SAT solvers to find such graphs, it becomes computationally infeasible to do so as the size of the graph increases. So, we take an analytical approach and introduce the technique of local sparsification on experiment graphs, using which we answer a crucial open question in experimental quantum optics, namely whether certain complex entangled quantum states can be constructed. This provides us with more insights into quantum resource theory, the limitation of specific quantum photonic systems and initiates the use of graph-theoretic techniques for designing quantum physics experiments.
构建大量量子光子实验的最有效自动化方法是通过具有特定属性的图的抽象表示。虽然人们利用人工智能和 SAT 解算器探索了寻找此类图的新方向，但随着图的大小增加，这样做在计算上变得不可行。因此，我们采用了一种分析方法，并引入了实验图局部稀疏化技术，从而回答了量子光学实验中的一个关键性开放问题，即某些复杂纠缠量子态是否可以构建。这让我们对量子资源理论、特定量子光子系统的局限性有了更多的了解，并开启了利用图论技术设计量子物理实验的先河。
{"title":"Graph-theoretic insights on the constructability of complex entangled states","authors":"L. Sunil Chandran, Rishikesh Gajjala","doi":"10.22331/q-2024-07-03-1396","DOIUrl":"https://doi.org/10.22331/q-2024-07-03-1396","url":null,"abstract":"The most efficient automated way to construct a large class of quantum photonic experiments is via abstract representation of graphs with certain properties. While new directions were explored using Artificial intelligence and SAT solvers to find such graphs, it becomes computationally infeasible to do so as the size of the graph increases. So, we take an analytical approach and introduce the technique of local sparsification on experiment graphs, using which we answer a crucial open question in experimental quantum optics, namely whether certain complex entangled quantum states can be constructed. This provides us with more insights into quantum resource theory, the limitation of specific quantum photonic systems and initiates the use of graph-theoretic techniques for designing quantum physics experiments.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":null,"pages":null},"PeriodicalIF":6.4,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141495979","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-02DOI: 10.22331/q-2024-07-02-1394
György P. Gehér, Campbell McLauchlan, Earl T. Campbell, Alexandra E. Moylett, Ophelia Crawford
We simulate the logical Hadamard gate in the surface code under a circuit-level noise model, compiling it to a physical circuit on square-grid connectivity hardware. Our paper is the first to do this for a logical unitary gate on a quantum error-correction code. We consider two proposals, both via patch-deformation: one that applies a transversal Hadamard gate (i.e. a domain wall through time) to interchange the logical $X$ and $Z$ strings, and another that applies a domain wall through space to achieve this interchange. We explain in detail why they perform the logical Hadamard gate by tracking how the stabilisers and the logical operators are transformed in each quantum error-correction round. We optimise the physical circuits and evaluate their logical failure probabilities, which we find to be comparable to those of a quantum memory experiment for the same number of quantum error-correction rounds. We present syndrome-extraction circuits that maintain the same effective distance under circuit-level noise as under phenomenological noise. We also explain how a $SWAP$-quantum error-correction round (required to return the patch to its initial position) can be compiled to only four two-qubit gate layers. This can be applied to more general scenarios and, as a byproduct, explains from first principles how the "stepping" circuits of the recent Google paper [1] can be constructed.
{"title":"Error-corrected Hadamard gate simulated at the circuit level","authors":"György P. Gehér, Campbell McLauchlan, Earl T. Campbell, Alexandra E. Moylett, Ophelia Crawford","doi":"10.22331/q-2024-07-02-1394","DOIUrl":"https://doi.org/10.22331/q-2024-07-02-1394","url":null,"abstract":"We simulate the logical Hadamard gate in the surface code under a circuit-level noise model, compiling it to a physical circuit on square-grid connectivity hardware. Our paper is the first to do this for a logical unitary gate on a quantum error-correction code. We consider two proposals, both via patch-deformation: one that applies a transversal Hadamard gate (i.e. a domain wall through time) to interchange the logical $X$ and $Z$ strings, and another that applies a domain wall through space to achieve this interchange. We explain in detail why they perform the logical Hadamard gate by tracking how the stabilisers and the logical operators are transformed in each quantum error-correction round. We optimise the physical circuits and evaluate their logical failure probabilities, which we find to be comparable to those of a quantum memory experiment for the same number of quantum error-correction rounds. We present syndrome-extraction circuits that maintain the same effective distance under circuit-level noise as under phenomenological noise. We also explain how a $SWAP$-quantum error-correction round (required to return the patch to its initial position) can be compiled to only four two-qubit gate layers. This can be applied to more general scenarios and, as a byproduct, explains from first principles how the \"stepping\" circuits of the recent Google paper [1] can be constructed.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":null,"pages":null},"PeriodicalIF":6.4,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141495954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-02DOI: 10.22331/q-2024-07-02-1391
Adam R. Brown
This paper proves the polynomial equivalence of a broad class of definitions of quantum computational complexity. We study right-invariant metrics on the unitary group—often called `complexity geometries' following the definition of quantum complexity proposed by Nielsen—and delineate the equivalence class of metrics that have the same computational power as quantum circuits. Within this universality class, any unitary that can be reached in one metric can be approximated in any other metric in the class with a slowdown that is at-worst polynomial in the length and number of qubits and inverse-polynomial in the permitted error. We describe the equivalence classes for two different kinds of error we might tolerate: Killing-distance error, and operator-norm error. All metrics in both equivalence classes are shown to have exponential diameter; all metrics in the operator-norm equivalence class are also shown to give an alternative definition of the quantum complexity class BQP.
My results extend those of Nielsen et al., who in 2006 proved that one particular metric is polynomially equivalent to quantum circuits. The Nielsen et al. metric is incredibly highly curved. I show that the greatly enlarged equivalence class established in this paper also includes metrics that have modest curvature. I argue that the modest curvature makes these metrics more amenable to the tools of differential geometry, and therefore makes them more promising starting points for Nielsen's program of using differential geometry to prove complexity lowerbounds.
In a previous paper my collaborators and I—inspired by the UV/IR decoupling that happens in the phenomenon of renormalization—conjectured that high- dimensional metrics that look very different at short scales will often nevertheless give rise at long scales to the same emergent effective geometry. The results of this paper provide evidence for those conjectures, since many complexity metrics that have radically different penalty factors and therefore radically different short- distance properties are shown to belong to the same long-distance equivalence class.
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