We introduce an approach for performing quantum state reconstruction on
systems of $n$ qubits using a machine-learning-bas ed reconstruction system
trained exclusively on $m$ qubits, where $m\geq n$. This approach removes the
necessity of exactly matching the dimensionality of a system under
consideration with the dimension of a model used for training. We demonstrate
our technique by performing quantum state reconstruction on randomly sampled
systems of one, two, and three qubits using machine-learning-bas ed methods
trained exclusively on systems containing at least one additional qubit. The
reconstruction time required for machine-learning-bas ed methods scales
significantly more favorably than the training time; hence this technique can
offer an overall savings of resources by leveraging a single neural network for
dimension-variable state reconstruction, obviating the need to train dedicated
machine-learning systems for each Hilbert space.
Dimension-adaptive machine-learning-bas ed quantum state reconstruction
次元適応型機械学習に基づく量子状態再構成
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Sanjaya Lohani,1, ∗ Sangita Regmi,1 Joseph M. Lukens,2
Sanjaya Lohani,1, ∗ Sangita Regmi,1 Joseph M. Lukens,2
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Ryan T. Glasser,3 Thomas A. Searles,1, † and Brian T. Kirby3, 4, ‡
Ryan T. Glasser,3 Thomas A. Searles,1, . and Brian T. Kirby3, 4, .
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1Dept. of Electrical & Computer Engineering, University of Illinois Chicago, Chicago, IL 60607, USA 2Quantum Information Science Section, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
1位。 イリノイ大学シカゴ校電気・計算機工学科, il 60607, usa 2quantum information science section, oak ridge national laboratory, oak ridge, tn 37831, usa 訳抜け防止モード: 1位。 シカゴ・イリノイ大学電気・計算機工学科, il 60607 usa 2quantum information science section, oak ridge national laboratory, oak ridge, tn 37831, usa
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3Tulane University, New Orleans, LA 70118, USA
3Tulane University, New Orleans, LA 70118, USA
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4DEVCOM Army Research Laboratory, Adelphi, MD 20783, USA
アメリカ、アデルフィ、MD 20783の4DEVCOM陸軍研究所
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(Dated: May 13, 2022)
(2022年5月13日廃止)
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We introduce an approach for performing quantum state reconstruction on systems of n qubits using a machine-learning-bas ed reconstruction system trained exclusively on m qubits, where m ≥ n.
m ≥ n の m 量子ビットのみを訓練した機械学習に基づく再構成システムを用いて,n 量子ビットシステム上で量子状態再構成を行う手法を提案する。 訳抜け防止モード: アプローチを導入する n量子ビット系の量子状態再構成を行う m ≥ n の m qubits のみにトレーニングされた,機械学習ベースの再構築システム。
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This approach removes the necessity of exactly matching the dimensionality of a system under consideration with the dimension of a model used for training.
We demonstrate our technique by performing quantum state reconstruction on randomly sampled systems of one, two, and three qubits using machine-learning-bas ed methods trained exclusively on systems containing at least one additional qubit.
The reconstruction time required for machine-learning-bas ed methods scales significantly more favorably than the training time; hence this technique can offer an overall savings of resources by leveraging a single neural network for dimension-variable state reconstruction, obviating the need to train dedicated machine-learning systems for each Hilbert space.
I. INTRODUCTION Estimating the properties of a quantum system through measurement is a task of fundamental importance in quantum information science.
私は... 導入 測定による量子システムの特性の推定は、量子情報科学において重要な課題である。
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Although methods exist for the partial characterization of quantum systems requiring relatively few measurements [1–6], the complete reconstruction of a density matrix has the distinct advantage of providing full information on any property of the system.
Complete reconstruction requires quantum state tomography (QST), where repeated measurements on an ensemble of identically prepared systems are used to estimate the system’s density matrix.
In general, QST consists of the preparation and experimental collection of measurement results and the purely classical and computational step of recovering the density matrix most consistent with the measurement results, known as quantum state reconstruction [7–9].
Various methods exist for performing quantum state reconstruction, including maximum likelihood estimation [8–14], Bayesian inference [6, 15–25], and machine-learning-bas ed techniques – supervised learning [26–42], semi-supervised learning [43–45], and reinforcement learning [46].
The exponential scaling of Hilbert space dimension with the number of qubits presents a challenge both experimentally and computationally for implementations of QST.
Hence, the number of distinct measurement bases desired will always scale exponentially.
したがって、異なる測定基準の数が常に指数関数的にスケールする。
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Similarly, the computational cost required to perform quantum state reconstruction using most leading techniques, such as maximum likelihood or Bayesian
estimation, also scales exponentially with system size.
推定はシステムサイズと指数関数的にスケールします
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While the resources required to perform the experimental measurements required for QST often eclipse the reconstruction time for small quantum systems (e g , one or two qubits), this is not necessarily the case for larger quantum systems [24, 47, 48].
For this reason, significant research has focused on developing alternative quantum state reconstruction methods with more favorable computational scaling.
One recently proposed approach for confronting quantum state reconstruction’s computational cost is to frontload the exponential scaling into the training period of a machine-learning-bas ed system [27, 28, 30].
In particular, recent results applying pre-trained networks to near-term intermediate scale quantum (NISQ) devices of up to four qubits revealed a significant increase in the training time as a function of the dimension of the underlying space, but only an extremely modest increase in the reconstruction time [30].
For example, after training, the reconstruction time was 0.77 ms for single-qubit systems, rising only to 0.8 ms for four qubits—a near-trivial increase considering the eight-fold growth in Hilbert space dimension.
例えば、訓練後の再構成時間は、シングルキュービットシステムでは 0.77 ms であり、4キュービットでは 0.8 ms まで上昇し、ヒルベルト空間次元の8倍の成長を考えると、ほぼ自明な増加である。 訳抜け防止モード: 例えば、トレーニング後の再建時間はシングルキュービットシステムで0.77msであった。 ヒルベルト空間次元の 8 倍の成長を考えると、四つの量子ビットに対して 0.8 ms にしか上昇しない。
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Note that the training of a network only needs to be performed once, and subsequently the network can be applied to any future datasets using comparatively modest resources.
For example, as described above, the pre-trained network for four-qubit systems can always perform full state reconstruction in 0.8 ms (on the hardware used in [30]) without any additional training.
Although machine-learning-bas ed reconstruction systems can be trained over the entire Hilbert space and used to reconstruct arbitrarily mixed states, the training generally focuses on a fixed Hilbert space dimension in order to limit the number of trainable parameters required to describe the system.
In other words, while it is in principle possible to train a network to accept variable-dimension input states, it requires a dramatic, and often practically infeasible, increase in network size
The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes.
DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/d ownloads/doe-public- access-plan).
doeは、doe public access plan(http://energy.g ov/downloads/doe-pub lic-access-plan)に従って、連邦政府が後援する研究の結果の公開アクセスを提供する。
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英語（論文から抽出）
日本語訳
スコア
2 FIG. 1. A schematic of our approach for dimension-adaptive quantum state reconstruction.
2 FIG.1。 次元適応量子状態再構成のための我々のアプローチの図式化
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The estimation of n-qubit quantum states uses a machine-learning-bas ed reconstruction system trained on m qubits, where m ≥ n.
n-qubit量子状態の推定には、m ≥ n で訓練された機械学習に基づく再構成システムを用いる。
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First, we append virtual results for m − n qubits via engineered padding (red dotted box) to n-qubit measurements (blue dotted box), and feed the padded measurements into a network pre-trained for m qubits.
At the output, the network returns an m-qubit density matrix, which is partially traced to return an estimate of the unknown quantum system of n qubits.
Therefore, an existing weakness of this approach is that a given trained network can only be applied to systems of precisely the dimension on which it was trained and does not generalize to smaller Hilbert spaces, instead requiring a separate system to be trained for every dimension.
Here we address this current limitation by proposing an approach for quantum state reconstruction on systems of n qubits using a machine-learning-bas ed system trained on m qubits, where m ≥ n.
ここでは、m ≥ n で訓練された機械学習に基づくシステムを用いて、n 量子ビット系の量子状態再構成のアプローチを提案する。
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We begin by generally relating the average reconstruction fidelity of an m-qubit quantum state to the average reconstruction fidelity of any of its reduced density matrices by applying the monotonicity of the fidelity.
We then extract this relationship specifically for ensembles of states randomly sampled according to the Hilbert–Schmidt (HS) measure for m ∈ {2, 3, 4}.
We interpret these results to indicate that reconstruction systems intended explicitly for m-qubit reconstruction implicitly inherit the capacity to perform n < m qubit reconstructions.
そこで本研究では,m-qubit再構成を意図した再構成システムが,n < m qubit再構成を行う能力を暗黙的に継承していることを示す。
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In addition, we describe a method, pictured in Fig 1, for augmenting n < m qubit tomography measurement results to m qubits such that the desired reconstruction can be obtained through the partial trace.
また、図1に示す方法として、n < m qubit のトモグラフィ測定結果を m qubit に拡大して、所望の再構成を部分的トレースで得られるようにする手法について述べる。
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Finally, we demonstrate our approach using simulated tomographic measurement data for n ≤ m qubits, complete inference utilizing networks trained with m ∈ {2, 3, 4}, and discuss the performance of our method.
最後に,m ∈ {2, 3, 4} で学習したネットワークを用いた完全推論法である n ≤ m qubits に対するシミュレーショントモグラフィ計測データを用いて,提案手法を実証し,本手法の性能について考察する。
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II. MACHINE-LEARNING-BAS ED QUANTUM
II。 機械学習に基づく量子
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STATE RECONSTRUCTION In this section, we describe the general details of our machine-learning-bas ed approach to m-qubit QST.
We implement a convolutional neural network (CNN) with a convolutional unit of kernel size (2, 2), strides of 1, ReLU as an activation function, and 25 filters.
The output of the CNN is fed into the next layer, which performs
CNNの出力は次のレイヤに供給され、実行されます。
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pooling with a pool-size (2, 2), followed by a second convolutional unit of the same configuration.
プールサイズ(2,2)でプーリングし、続いて同じ構成の第2の畳み込みユニットを使用する。
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Then, we combine two dense layers, followed by a dropout layer with a rate of 0.5, which is then attached to an output layer predicting τ -vectors (Cholesky coefficients of the density matrix [8]).
The mean square loss between the target and predicted τ is evaluated and fed back to optimize the network’s trainable parameters using the Adagrad optimizer.
At the output layer is attached a pipeline that rearranges the predicted τ -vectors into density matrices.
出力層には、予測されたτ-ベクトルを密度行列に並べ替えるパイプラインが取り付けられている。
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The pipeline is built into the same graph of the network for the purposes of computing the average fidelity per epoch for crossvalidation and outputting the density matrix directly to avoid post-processing.
Note that the physicality of ˜ρ is guaranteed through the Cholesky decomposition, which ensures positive semidefiniteness [8, 12].
シュρの物理性はコレスキー分解によって保証され、正の半定性 [8, 12] が保証される。
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Finally, at the end of the network, the fidelity F between the predicted density matrix ˜ρ and the
最後に、ネットワークの最後に、予測された密度行列 ρ とネットワークの間の忠実度 f が現れる。
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target ρ is evaluated as F =
ターゲット ρ は F = として評価される
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˜ρρ ˜ρ . An in-depth
˜ρρ ˜ρ . 奥行き
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description of the network architecture is given in [30].
ネットワークアーキテクチャの説明は[30]に書かれています。
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Previous work suggests an approximately exponential separation in the computational resources required to train a network of the type described above compared to using it for reconstruction.
For example, an analysis of the explicit training and inference times for systems of one to four qubits showed that, using the same computational resources, the training of a one-qubit network took 123 s but only 0.77 ms to perform reconstruction,
compared to 1380 s and 0.80 ms, respectively, for a fourqubit system [30].
それぞれ 1380 s と 0.80 s の 4 ビットシステム [30] と比較してみましょう
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The modest scaling in inference times is the main appeal of machine-learning-bas ed reconstruction methods but comes at the cost of an intensive upfront training period.
Such unfavorable scaling in the training times of neural networks are reminiscent of those found in reconstruction approaches based on maximum likelihood [47, 48] or Bayesian estimation [22, 24]; however, the network has the advantage that these resources can be expended ahead of time and only once.
Here we seek to further mitigate the overhead required for training by repurposing a network trained on systems of a particular dimension for inference of all lowerdimensional systems as well.
The general approach is pictured in Fig 1 and described in detail in Sec.
一般的なアプローチは図1で描かれており、secで詳細に説明されている。
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IV. In order to illustrate the proof of concept and train and test our reconstruction approach, we use mixed quantum states sampled according to the HS measure.
The choice of sampling according to the HS measure as opposed to others is due to the unique property that it induces a flat Euclidean geometry into the mixed states [49, 50] and has hence found wide adoption in various studies of quantum states.
We note, however, that many other distributions of random quantum states exist and have been studied in various contexts, including as prior distributions for Bayesian inference [19, 21, 22] and training sets for machine-learning-bas ed reconstruction [31].
(For completeness, in Appendix A we reproduce the results of this manuscript using density matrices sampled according to the Bures metric, another distribution of longstanding significance in quantum information [49, 51].)
We simulate the associated 6m Pauli measurement outcomes for systems with m ∈ {2, 3, 4} qubits directly from expectation values, which physically corresponds to the infinite-measurement limit (i.e., no statistical noise).
予想値から直接 m ∈ {2, 3, 4} の量子ビットを持つ系の関連する 6m Pauli の測定結果をシミュレートする。 訳抜け防止モード: m ∈ { 2 の系に対する関連する 6 m Pauli の測定結果をシミュレートする。 3, 4 } 期待値から直接量子ビット。 物理的には無限の測定限界(統計ノイズなし)に対応する。
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For each scenario, we split the sampled data into a training set comprising 35,000 states and a validation set of 500 states to cross-validate the network performance per epoch.
After training, we generate test sets that are entirely unknown to the trained network.
トレーニング後、トレーニングされたネットワークに完全に未知のテストセットを生成します。
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The code to generate all datasets can be found in [52].
すべてのデータセットを生成するコードは [52] にある。
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III. REDUCED DENSITY MATRIX FIDELITY
III。 還元密度行列係数
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Machine-learning-bas ed techniques for quantum state reconstruction have been applied to systems of a variety of dimensions.
量子状態再構成のための機械学習技術は、様々な次元のシステムに適用されている。
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Despite attaining high average reconstruction fidelity for the overall state, to our knowledge the way in which this translates to the fidelity of the reduced density matrices has not been considered.
Such an analysis is useful if, after tomography and reconstruction, study of a specific subspace is desired without performing additional reconstruction.
A relationship between the fidelity of two density matrices and the fidelity between any of their corresponding
2つの密度行列の忠実度と対応するいずれかの忠実度の関係
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3 reduced density matrices follows immediately from the well-known property of monotonicity [53, 54].
3 還元密度行列は単調[53, 54]のよく知られた性質から直ちに従う。
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In particular, the fidelity F (ρAB, σAB) between any two density matrices ρAB and σAB, and corresponding reduced density matrices ρA = TrB(ρAB) and σA = TrB(σAB) is bounded by F (ρAB, σAB) ≤ F (ρA, σA) where A and B denote arbitrary bipartitions of each state.
特に、2つの密度行列 ρAB と σAB の間の忠実度 F (ρAB, σAB) と対応する還元密度行列 ρA = TrB(ρAB) と σA = TrB(σAB) は F (ρAB, σAB) ≤ F (ρA, σA) で有界である。
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In the context of quantum state reconstruction, we can consider ρAB as the actual ground truth state and σAB as the reconstruction.
量子状態再構成の文脈では、ρABを実際の基底真理状態、σABを再構成と考えることができる。
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Hence, the average reconstruction fidelity for any corresponding reduced density matrices over n qubits of an m-qubit reconstruction is lower bounded by the average m-qubit reconstruction fidelity.
Note that we are only able to apply the monotonicity of the partial trace to reconstruction methods that guarantee the physicality of the final density matrix, such as described in Sec.
It is worth emphasizing that the monotonicity of the fidelity only applies to specific pairs of density matrices.
忠実度の単調性は特定の密度行列のペアにのみ適用されることを強調する価値がある。
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Yet in the context of machine-learning-bas ed tomography, we are interested primarily in averages over distributions of quantum states, as we seek to establish bounds on tomographic performance that would apply to a variety of initially unknown density matrices.
And as discovered previously, the average performance of machinelearning-base d reconstruction techniques can be heavily dependent on the distribution of density matrices used to calculate the average [31].
Therefore, we stress that the mean reconstruction fidelity obtained during training only bounds the average n < m reduced density matrix reconstruction fidelities (through monotonicity) when the test states are drawn from the same distribution, or more precisely, when the distribution of n-qubit test states is equal to the distribution resulting from tracing out m − n qubits from the m-qubit training distribution.
したがって、訓練中に得られた平均再構成忠実度は、試験状態が同じ分布から引き出される場合、またはより正確には、m-量子ビットトレーニング分布からm − n量子ビットをトレースして得られる分布と等しい場合、平均n < m 還元密度行列再構成フィディリティ(単調性による)にのみ制限されることを強調した。
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In other words, we cannot use monotonicity to obtain a completely general lower bound only based on network performance during training, as the averages depend on the distribution from which the states are drawn during deployment.
The development of custom distributions of random quantum states that mimic various general features of quantum systems could potentially limit the impact of mismatched training and test distributions in practice [32].
For illustrative purposes we now perform numerical simulations to compare the actual average reconstruction fidelity of the reduced density matrices to the lower bound determined by the monotonicity, using the methods described in Sec.
The average reconstruction fidelity of each network was determined, which as described above, should serve as the lower bound on the average reconstruction fidelity of the
III: the reduced density matrices of a state reconstructed using machine-learning-bas ed methods maintain or, as in Fig. 2, improve fidelity in comparison with the reconstruction of the entire state.
If, at this stage, we were aware that we were restricted to an m-qubit reconstruction technique we could physically augment the n qubit target state ρn with an arbitrary system of m − n single qubit states σ to create the state
この段階で、我々はm-qubit再構成技術に制限されていることを知っていれば、n qubitターゲット状態 ρn を m − n 個の単一量子状態 σ の任意の系で物理的に拡張して状態を生成することができる。
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ρm = ρn ⊗ σn+1 ⊗ σn+2 ⊗ ... ⊗ σm−n.
ρm = ρn , σn+1 , σn+2 , ... , σm−n である。
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(2) We could then collect standard tomographic measurement results for the total ρm system, perform reconstruction and obtain ˜ρm, the reconstruction of ρm.
(2) そこで,全ρm系の標準トモグラフィー測定結果を収集し,再構成を行い,ρmの再構成を行う。
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Then, to obtain the desired result, the reconstruction of the ρn state, we perform a partial trace ˜ρn = Trσ(˜ρm) where Trσ indicates tracing over the added single qubits.
However, not only is physically augmenting a quantum system experimentally challenging, but it is also unnecessary.
しかし、物理的に量子システムを実験的に増強するだけでなく、それは不要である。
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As pictured in Fig 1, tomographic data for ρn alone can be augmented with synthetic data for the added qubits, meaning that all modifications required to use an m-qubit reconstruction technique on an n < m qubit system can be performed in postprocessing.
図1に示すように、ρn 用のトモグラフィーデータは、追加された qubit の合成データで拡張することができ、n < m qubit システム上で m-qubit 再構成技術を使用するために必要な全ての修正は、後処理で実行できる。
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While in principle we could create synthetic measurement results that place the added single-qubit states σk in any arbitrary state, it is conceptually simple to make them all completely mixed.
The benefit of using separable and completely mixed single-qubit states is twofold.
分離可能で完全に混合されたシングルキュービット状態を使用することの利点は2倍である。
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First, with each additional state being separable the joint measurement results are classical products of individual measurement results.
第一に、各追加状態が分離可能であるため、ジョイント測定結果は個々の測定結果の古典的産物である。
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Further, the completely mixed nature of the added states means that measurement outcomes are equal and exactly 1/2 for all projective measurements in all orientations.
Therefore, postprocessing tomography data for ρn to ρm is merely a matter of multiplying each of the original measurement results by (1/2)m−n for a standard overcomplete basis consisting of projections on each Pauli eigenvector.
The results of following this procedure using networks with m ∈ {2, 3, 4} to reconstruct states with n < m are shown in Fig 3.
m ∈ {2, 3, 4} のネットワークを用いて n < m の状態を再構成するこの手順に従う結果は、図 3 に示される。
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Given the expressive power of machine-learning-bas ed reconstruction techniques it is reasonable to question if it is even necessary to perform the synthetic basis extension in order perform reconstruction with lower dimensional states.
For example, a naive alternative would be to merely zero pad all missing measurement results and use this as the input to the network, especially as the network is itself constrained to always produce a physi-
FIG. 2. Reconstruction fidelity versus subsystem of predicted density matrix.
FIG.2。 予測密度行列の再構成忠実性とサブシステム
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For example, when ρ represents a four-qubit system, then the subsystems Tr0(ρ), Tr01(ρ), Tr012(ρ) represent a three-qubit, two-qubit, and one-qubit quantum system, respectively.
reduced density matrices, and is plotted in Fig 2 as the horizontal dashed lines.
密度行列が減少し、水平破断線として図2でプロットされる。
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We then use each network to reconstruct another, independently and generally different, ensemble of random quantum states sampled according to the HS metric on m qubits, and for each reconstruction also perform a local trace for every decrement of one qubit and calculate the fidelity against the ground truth state.
As evident in Fig 2, in all cases the average fidelity outperforms the lower bound found from the monotonicity.
図2で明らかなように、すべての場合において平均忠実度は単調性から得られる下界よりも優れる。
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Intuitively, the high fidelity of the reduced density matrices suggests that any m-dimensional reconstruction method implicitly includes some ability to reconstruct n < m dimensional systems as well, provided it can be harnessed in a consistent fashion.
直観的には、還元密度行列の高忠実性は、任意の m 次元の再構成法に暗黙的に n < m 次元の系を再構築する能力も含んでいることを示唆している。
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This observation forms the inspiration for the general dimension-adaptive reconstruction scheme described in detail below.
この観察は、後述の一般次元適応型再構成法に着想を与えている。
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IV. EXTENDING SYSTEM DIMENSION WITH
IV。 システム次元の拡張
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SYNTHETIC MEASUREMENT RESULTS In the previous section, inference was performed on true m-qubit states using m-qubit-trained neural networks.
By tracing down these larger m-qubit states postinference, states with n < m qubits were obtained leading to fidelities between the inferred and ground truth subsystems that increased steadily—a finding in agreement with expectations from monotonicity.
これらの大きな m-量子ビット状態の追従により、n < m 量子ビットを持つ状態が得られ、推論された真理サブと基底真理サブのフィディティが徐々に増大し、モノトニック性からの期待と一致した。 訳抜け防止モード: これらの大きなm - qubit状態のポスト推論を辿ることで、n < m qubits の状態が得られた。 単調性からの期待と一致して 着実に増加しました
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Now we look to build upon these ideas and address the more challenging and unexplored situation where the target quantum system consists of n qubits and one has access to a reconstruction apparatus designed only for m > n qubits, thus requiring some method to bridge the mismatched
5 genta line shows the average fidelity of two states chosen at random from the HS distribution, and the olive line shows when one is always the identity.
2 and 3, it is certainly tempting to take the average fidelities found for n = m as lower bounds for the n < m cases.
2 と 3 は、n < m の場合の下限として n = m で見つかる平均的フィデリティを取るのが誘惑的である。
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Yet whereas monotonicity justifies such a bound in Sec.
一方、単調性は Sec においてそのような境界を正当化する。
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III, it does not apply to the results in Fig 3.
第三に、Fig 3の結果には当てはまらない。
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This can be understood through examination of Eq (2).
これは Eq (2) の検証によって理解することができる。
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Even if the n-qubit target states ρn are HS-distributed in 2n dimensions, the mqubit extension ρm is not HS-distributed in 2m: m− n of its qubits are restricted to fixed states.
Thus, any training results obtained on m-qubit HS-distributed quantum states cannot be used to bound the fidelities of ρm defined in Eq (2), which would have been required to thereafter bound the traced-down versions ρn.
Nevertheless, despite the formal inapplicability of monotonicity, the observed scaling does match our initial intuition motivated by it: a single neural network is able to infer quantum states from Hilbert spaces of lower dimension than that on which it is trained, with fidelity even higher than the designed high-dimensional case.
In this work, we have proposed a physically motivated approach to performing quantum state reconstruction on systems of n qubits when restricted to a state reconstruction technique intended for m ≥ n qubits.
本研究では,m ≥ n 量子ビットを対象とする状態再構成技術に制限された場合,n 量子ビット系の量子状態再構成を物理的に動機づける手法を提案する。
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The utility of this approach is based on previous results indicating an approximately exponential separation in the required resources for training a neural network to perform quantum state reconstruction and reconstruction itself.
Hence, efforts to avoid training an individual network for every potential system dimension that may be encountered in experimental scenarios can potentially offer significant resource savings.
We began by describing a close link between the average reconstruction fidelity of m-qubit states and their n < m qubit reduced density matrices using the wellknown monotonicity bound of the fidelity.
まず,m-qubit 状態の平均再構成忠実度と,その n < m qubit 還元密度行列との密接な関係を,よく知られた単調性境界を用いて記述した。
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In particular, the average reconstruction fidelity of an m-qubit QST approach found during training can serve as the lower bound on the average reconstruction fidelity of any reduced density matrix from such an m-qubit reconstruction.
As a proof of principle, we included an illustrative example based on simulated quantum state tomography measurements using a pre-trained machine-learningbase d state reconstruction system for m ∈ {2, 3, 4}.
We performed reconstruction for all m-qubit systems for each of the pre-trained networks and compared their average m-qubit performance to the fidelity of their reduced density matrices.
The magenta line shows the average fidelity between two random density matrices sampled from the Hilbert-Schmidt (HS) measure, whereas the olive line represents the average fidelity between a maximally mixed state and a random density matrix sampled from the HS measure for a system of n qubits.
cal state. More specifically, given an m-qubit network designed to take 6m measurement outcomes as input, we could only fill the first 6n measurement results zeroing out the remaining 6m − 6n elements.
This approach is not well motivated physically, as measurement inputs for the m-qubit system are joint measurements that include qubits unavailable to the n-qubit system.
However, zero padding is nevertheless a straightforward way to perform n-qubit reconstruction with a network trained on m qubits and is surprisingly effective when used to replace missing measurements for an n = m qubit reconstruction [27].
しかしながら、ゼロパディングは、m qubits でトレーニングされたネットワークで n 量子ビットの再構成を行うための簡単な方法であり、n = m qubit の再構成で失われた測定値を置き換えるのに驚くほど効果的である [27]。 訳抜け防止モード: しかし ゼロパディングは m 量子ビットを訓練したネットワークを用いて n-量子ビット再構成を行う 驚くほど効果的です n = m qubit のリコンストラクションで失われた測定値を置き換えるために使われる[27 ]。
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Note that the order of the 6n measurement results is such that the local bases correctly match their portion of the m-qubit joint measurements.
To demonstrate the dramatic difference between this naive zero-padding approach and the basis augmentation approach above we have included the blue lines in Fig. 3.
The separation in average fidelity between these two approaches is significant.
これら2つのアプローチ間の平均忠実度の分離は重要である。
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We compare the zero-padding approach to the trivial strategies of
ゼロパディングアプローチと自明な戦略を比較します
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(i) selecting another n-qubit state at random from the same distribution or
(i)同じ分布からランダムに別のnビット状態を選択するか
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(ii) selecting the maximally mixed state always.
(ii) 常に最大混合状態を選択すること。
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Interestingly, we find that zero padding only performs marginally better than randomly selecting another state according to the HS measure and performs worse than always selecting identity.
These results are shown visually in Fig 3 where the ma-
これらの結果は図3に表示されます。
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英語（論文から抽出）
日本語訳
スコア
struction of reduced density matrices, both implied by the monotonicity bound and confirmed in our numerical results, we proposed a method for leveraging m-qubit QST systems to perform n < m qubit reconstructions.
単調性に拘束された密度行列の縮小構造を数値計算により検証し, n < m qubit の再構成を行うために m-qubit qst システムを活用する方法を提案した。
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Our approach consists of expanding n qubits to m > n qubit systems via postprocessing and then recovering the n-qubit density matrix through partial trace.
In particular, we propose augmenting the collected tomography data with results from fictitious single-qubit states.
特に,収集したトモグラフィーデータを,架空の単一量子状態の結果で拡張することを提案する。
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In our study, we have opted to use completely mixed singlequbit states to achieve this due to their isotropic behavior under projective measurement and the relative simplicity of how these states alter joint measurements, i.e., as a multiplicative factor.
We demonstrate the proof of principle of this approach using systems of up to four qubits.
我々は、最大4キュービットのシステムを用いて、このアプローチの原理の証明を示す。
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Further, we compare the performance of our technique with the naive approach of expanding the dimensions through zero padding.
さらに,本手法の性能を,ゼロパディングにより次元を拡大するナイーブアプローチと比較した。
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The zero-padding method performs significantly worse than our simulated measurement approach and only marginally better than the theoretical lower bound of sidestepping tomography and randomly guessing an answer.
While we limited our discussion to systems based on qubits and collections of qubits, restricting the possible Hilbert space dimensions to powers of two, extensions to arbitrary dimensions are straightforward.
Further, based on previous results showing the impact of engineering training sets to emphasize specific system features [31, 32], further improvements could potentially be found by developing training sets that explicitly consider the distribution of their reduced density matrices.
ACKNOWLEDGMENTS Work by S. Lohani and T. A. Searles was supported in part by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under contract number DE-SC0012704.
裏書き S. Lohani と T. A. Searles の業績は、契約番号 DE-SC0012704 の下で、アメリカ合衆国エネルギー省、科学省、国家量子情報科学研究センター、C2QA の共同設計センターによって支援された。
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A portion of this work was performed at Oak Ridge National Laboratory, operated by UT-Battelle for the U.S. Department of Energy under contract no.
この研究の一部はオークリッジ国立研究所 (Oak Ridge National Laboratory) で行われ、UT-Battelle が契約No.の下でアメリカ合衆国エネルギー省のために行った。
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DE-AC05-00OR22725.
DE-AC05-00OR22725。
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J. M. Lukens acknowledges funding by the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research, through the Early Career Research Program (Field Work Proposal ERKJ353).
The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government.
Additionally, this material is based upon work supported by, or in part by, the Army Research Laboratory and the Army Research Office under contract/grant numbers W911NF-19-2-0087 and W911NF-20-2-0168.
distribution In order to illustrate the concept for other cases, we sample 35,500 random quantum states ρ according to the Bures measure for the given m [51].
流通 他のケースの概念を説明するために、与えられた m [51] のバーズ測度に従って35,500個のランダム量子状態 ρ をサンプリングする。
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Similarly, we also simulate the associated 6m Pauli measurement outcomes for systems with m ∈ {2, 3, 4} qubits directly from expectation values.
同様に、m ∈ {2, 3, 4} キュービットを持つ系の関連する 6m Pauli の測定結果を期待値から直接シミュレートする。
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As described in the main text, we split the sampled data into a training set of size 35,000 and a validation set of size 500 to cross-validate the network performance per epoch.
We implement a batch size of 100 in the training of a network.
ネットワークのトレーニングにおいて,バッチサイズ100を実装した。
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After training, we generate test sets, again, using the Bures measure for the same and lower qubit systems that are entirely unknown to the trained network.
Finally, the reconstruction fidelity with respect to subsystem size and number of qubits are, respectively, shown in Fig 4
最後に、サブシステムサイズとキュービット数に対する再構成忠実度を図4に示す。
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(a) and (b). Although the average reconstruction fidelities for states sampled according to the Bures metric are slightly lower than the those drawn from the HS metric [see Fig 2 in the main text], the same important scaling trends hold.
quantum states For completeness we include here the expression for the average fidelity (cid:104)F(cid:105)N between two random mixed states of dimension N generated according to the HS measure.
量子状態 完全性について、HS測度によって生成される2つの非乱混合次元 N の間の平均忠実度 (cid:104)F(cid:105)N の式を含める。
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We take our results from [55] where a more general expression applicable to two random mixed states chosen according to an arbitrary induced measure is presented.
Using these expressions we find for one, two, and three qubit states, respectively, that (cid:104)F(cid:105)2 = 0.67, (cid:104)F(cid:105)4 = 0.59, and (cid:104)F(cid:105)8 = 0.57.
In addition to the average fidelity between two random density matrices chosen according to the HS measure, we also show three other average fidelities in Figs.
Figure 4(b) includes the average fidelity between two random quantum states chosen according to the Bures measure.
図4(b)は、bures測度に従って選択された2つのランダム量子状態の平均忠実性を含む。
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In [55], an analytical result is given for this situation in the case of single-qubit states: (cid:104)F(cid:105)2 = 0.590, which we supplement with numerical results for two and three qubits to create the relevant curve in Fig 4b.
7 FIG. 4. Test and train with random quantum states sampled from the Bures metric.
7 図4。 ビューズ計量からサンプリングされたランダム量子状態のテストと訓練。
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(a) Reconstruction fidelity with respect to subsystem of predicted quantum states.
(a)予測量子状態のサブシステムに対する再構成忠実性。
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(b) Reconstruction fidelity versus number of qubits.
b) 量子ビット数に対する再構成の忠実度
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[1] A. K. Ekert, C. M. Alves, D. K. Oi, M. Horodecki, and L. C. Kwek, Phys.
[1] A. K. Ekert, C. M. Alves, D. K. Oi, M. Horodecki, L. C. Kwek, Phys。
0.43
Rev. Lett.
Rev. Lett.
0.43
88, P. Horodecki, 217901 (2002).
88, P. Horodecki, 217901 (2002)。
0.66
[2] S. T. Flammia and Y.
[2] S. T. Flammia, Y。
0.42
-K. Liu, Phys.
-K。 Liu, Phys。
0.42
Rev. Lett.
Rev. Lett.
0.43
106, 230501 (2011).
106, 230501 (2011).
0.43
[3] C. Spengler, M. Huber, S. Brierley, T. Adaktylos, and
[3] c. spengler, m. huber, s. brierley, t. adaktylos, and
0.40
[9] D. F. James, P. G. Kwiat, W. J. Munro, and A. G. White, in Asymptotic Theory of Quantum Statistical Inference: Selected Papers (World Scientific, 2005) pp. 509– 538.
D. F. James, P. G. Kwiat, W. J. Munro, A. G. White, in Asymptotic Theory of Quantum Statistical Inference: Selected Papers (World Scientific, 2005) pp. 509–538。
0.45
[10] Z. Hradil, Phys.
9] Z. Hradil, Phys.
0.37
Rev. A 55, R1561 (1997).
a 55, r1561 (1997) を参照。
0.68
[11] K. Banaszek, G. M. D’Ariano, M. G. A. Paris, and M. F.
K. Banaszek, G. M. D'Ariano, M. G. A. Paris, M. F.
0.43
B. C. Hiesmayr, Phys.
B. C. Hiesmayr, Phys
0.45
Rev. A 86, 022311 (2012).
86, 022311 (2012)を参照。
0.61
Sacchi, Phys.
Sacchi, Phys。
0.40
Rev. A 61, 010304 (1999).
61, 010304 (1999) を参照。
0.57
[4] H. -Y. Huang, R. Kueng, and J. Preskill, Nat.
[4]H。 -y。 huang、r. kueng、j. preskill、nat。
0.41
Phys. 16, [12] D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G.
Phys 16, D. F. V. James, P. G. Kwiat, W. J. Munro, A. G.
0.38
1050 (2020).
1050 (2020).
0.43
[5] J. Eisert, D. Hangleiter, N. Walk, I. Roth, D. Markham, R. Parekh, U. Chabaud, and E. Kashefi, Nat.
J. Eisert, D. Hangleiter, N. Walk, I. Roth, D. Markham, R. Parekh, U. Chabaud, E. Kashefi, Nat. 訳抜け防止モード: 【5】j.アイサート、d.ハングライト、n.ウォーク、 i. roth, d. markham, r. parekh, u. chabaud, e. kashefi、nat。
0.60
Rev. Phys. 2, 382 (2020).
Phys博士。 2, 382 (2020).
0.54
White, Phys. Rev. A 64, 052312 (2001).
白、白。 a 64, 052312 (2001) を参照。
0.48
[13] A. I. Lvovsky, J. Opt.
[13] A. I. Lvovsky, J. Opt.
0.46
B: Quantum Semiclass.
b: 量子半クラス。
0.88
Opt. 6, S556 (2004).
オプト 6, S556(2004年)。
0.49
[14] J. A. Smolin, J. M. Gambetta, and G. Smith, Phys.
J. A. Smolin, J. M. Gambetta, G. Smith, Phys.
0.38
Rev. [6] J. M. Lukens, K. J. Law, and R. S. Bennink, npj Quan-
レヴ J. M. Lukens, K. J. Law, R. S. Bennink, npj Quan
0.34
Lett. 108, 070502 (2012).
Lett! 108, 070502 (2012).
0.37
tum Inf. 7, 113 (2021).
inf所属。 7, 113 (2021).
0.72
[7] R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro,
[7]R.T.Thew,K.Nemoto,A. G.White,W.J.Munro
0.37
[15] R. Blume-Kohout, New J. Phys.
[15]R. Blume-Kohout, New J. Phys
0.43
12, 043034 (2010).
12, 043034 (2010).
0.43
[16] F. Husz´ar and N. M. T. Houlsby, Phys.
F. Husz ́ar と N. M. T. Houlsby, Phys.
0.43
Rev. A 85, 052120
85, 052120条
0.24
Phys. Rev. A 66, 012303 (2002).
Phys 66, 012303 (2002)を参照。
0.44
(2012). [8] J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat, Adv.
(2012). J. B. Altepeter, E. R. Jeffrey, P. G. Kwiat, Adv.
0.41
At. Mol. Opt.
です。 モル オプト
0.42
Phys. 52, 105 (2005).
Phys 52, 105 (2005).
0.35
[17] K. S. Kravtsov, S. S. Straupe, I. V. Radchenko, N. M. T. Houlsby, F. Husz´ar, and S. P. Kulik, Phys.
K. S. Kravtsov, S. S. Straupe, I. V. Radchenko, N. M. T. Houlsby, F. Husz ́ar, S. P. Kulik, Phys.
0.43
Rev. A 87,
87号室長。
0.38
英語（論文から抽出）
日本語訳
スコア
8 062122 (2013).
8 062122 (2013).
0.42
[18] Y. -L.
[18] Y。 -L。
0.40
Seah, J. Shang, H. K. Ng, D. J. Nott, and B.
Seah, J. Shang, H. K. Ng, D. J. Nott, B。
0.92
-G. Englert, New J. Phys.
-G。 ニュー・J・フィス(New J. Phys)。
0.47
17, 043018 (2015).
17, 043018 (2015).
0.43
[19] C. Granade, J. Combes, and D. G. Cory, New J. Phys.
C. Granade, J. Combes, D. G. Cory, New J. Phys.
0.40
18, 033024 (2016).
18, 033024 (2016).
0.43
[20] B. P. Williams and P. Lougovski, New J. Phys.
20] b. p. williams and p. lougovski, new j. phys.
0.42
19, 043003 (2017).
19, 043003 (2017).
0.42
[37] ´E. Genois, J. A. Gross, A. Di Paolo, N. J. Stevenson, G. Koolstra, A. Hashim, I. Siddiqi, and A. Blais, PRX Quantum 2, 040355 (2021).
37年。 Genois, J. A. Gross, A. Di Paolo, N. J. Stevenson, G. Koolstra, A. Hashim, I. Siddiqi, A. Blais, PRX Quantum 2, 040355 (2021)。 訳抜け防止モード: 37年。 genois, j. a. gross, a. di paolo, n. j. stevenson g. koolstra、a. hashim、i. siddiqi、a. blais。 prx量子2,040355 (2021) である。
0.50
[38] Y. S. Teo, S. Shin, H. Jeong, Y. Kim, Y.
[38]Y.S.Teo,S.Shin,H.Jon g,Y.Kim,Y。
0.74
-H. Kim, G. I. Struchalin, E. V. Kovlakov, S. S. Straupe, S. P. Kulik, G. Leuchs, and L. L. S´anchez-Soto, New J. Phys.
-h。 Kim, G. I. Struchalin, E. V. Kovlakov, S. S. Straupe, S. P. Kulik, G. Leuchs, L. L. S ́anchez-Soto, New J. Phys。
0.57
23, 103021 (2021).
23, 103021 (2021).
0.43
[21] T. T. Mai and P. Alquier, J. Stat.
[21]T.T. MaiとP. Alquier, J. Stat
0.45
Plan. Inference 184,
計画。 推理184年。
0.58
[39] E. S. Tiunov, V. Tiunova, A. E. Ulanov, A. Lvovsky, and
39] E. S. Tiunov, V. Tiunova, A. E. Ulanov, A. Lvovsky,
0.48
62 (2017).
62 (2017).
0.42
[22] J. M. Lukens, K. J. H. Law, A. Jasra, and P. Lougovski,
J. M. Lukens, K. J. H. Law, A. Jasra, P. Lougovski
0.39
New J. Phys.
新しいJ. Phys。
0.89
22, 063038 (2020).
22, 063038 (2020).
0.42
[23] E. M. Simmerman, H.
He23] E. M. Simmerman, H.
0.46
-H. Lu, A. M. Weiner, and J. M.
-h。 Lu, A.M. Weiner, J.M.
0.60
Lukens, Opt.
Lukens, Opt
0.30
Lett. 45, 2886 (2020).
Lett! 45, 2886 (2020).
0.37
[24] H. -H.
[24]H。 -h。
0.56
Lu, K. V. Myilswamy, R. S. Bennink, S. Seshadri, M. S. Alshaykh, J. Liu, T. J. Kippenberg, D. E. Leaird, A. M. Weiner, and J. M. Lukens, arXiv:2108.04124 (2021).
Lu, K. V. Myilswamy, R. S. Bennink, S. Seshadri, M. S. Alshaykh, J. Liu, T. J. Kippenberg, D. E. Leaird, A. M. Weiner, J. M. Lukens, arXiv:2108.04124 (2021)。
0.40
[25] J. C. Chapman, J. M. Lukens, B. Qi, R. C. Pooser, and
J. C. Chapman, J. M. Lukens, B. Qi, R. C. Pooser, そして
0.89
N. A. Peters, Opt.
N.A.ピータース、オプト。
0.62
Express 30, 15184 (2022).
15184号機(2022年)。
0.49
[26] S. Lu, S. Huang, K. Li, J. Li, J. Chen, D. Lu, Z. Ji, Y. Shen, D. Zhou, and B. Zeng, Phys.
[26]S. Lu, S. Huang, K. Li, J. Li, J. Chen, D. Lu, Z. Ji, Y. Shen, D. Zhou, B. Zeng, Phys.
0.47
Rev. A 98, 012315 (2018).
a98, 012315 (2018)を参照。
0.59
A. Fedorov, Optica 7, 448 (2020).
A. Fedorov, Optica 7, 448 (2020)。
0.86
[40] A. M. Palmieri, E. Kovlakov, F. Bianchi, D. Yudin, S. Straupe, J. D. Biamonte, and S. Kulik, npj Quantum Inf.
A.M. Palmieri, E. Kovlakov, F. Bianchi, D. Yudin, S. Straupe, J. D. Biamonte, S. Kulik, npj Quantum Inf.
0.46
6, 20 (2020).
6, 20 (2020).
0.42
[41] M. Neugebauer, L. Fischer, A. J¨ager, S. Czischek, S. Jochim, M. Weidem¨uller, and M. G¨arttner, Phys.
He41] M. Neugebauer, L. Fischer, A. J sager, S. Czischek, S. Jochim, M. Weidem suller, M. G sarttner, Phys。 訳抜け防止モード: ][41]m.ノイゲバウアー,l.フィッシャー,a.j.ジャガー, s. czischek, s. jochim, m. weidem suller, m. g sarttner, phys。
0.55
Rev. A 102, 042604 (2020).
A 102, 042604 (2020)。
0.58
[42] R. Wang, C. Hernani-Morales, J. D. Mart´ın-Guerrero, E. Solano, and F. Albarr´an-Arriagada, Quantum Sci.
[42] R. Wang, C. Hernani-Morales, J. D. Mart ́ın-Guerrero, E. Solano, F. Albarr ́an-Arriagada, Quantum Sci
0.36
Technol. 7, 015010 (2021).
テクノル 7, 015010 (2021).
0.46
[43] S. Ahmed, C. S. Mu˜noz, F. Nori, and A. F. Kockum,
43] S. Ahmed, C. S. Mu snoz, F. Nori, A. F. Kockum
0.45
Phys. Rev. Lett.
Phys Rev. Lett.
0.35
127, 140502 (2021).
127, 140502 (2021).
0.42
[44] J. Carrasquilla, G. Torlai, R. G. Melko, and L. Aolita,
[44]J. Carrasquilla,G. Torlai,R. G. Melko,L. Aolita
0.44
Nat. Mach. Intell.
Nat! マッハ インテリ。
0.38
1, 155 (2019).
1, 155 (2019).
0.43
[27] S. Lohani, B. T. Kirby, M. Brodsky, O. Danaci, and R. T.
[27]S. Lohani, B. T. Kirby, M. Brodsky, O. Danaci, R. T.
0.46
[45] S. Lohani, E. M. Knutson, and R. T. Glasser, Commu-
[45]S. Lohani, E. M. Knutson, R. T. Glasser, Commu
0.44
Glasser, Mach.
グラスラー、マッハ。
0.58
Learn. : Sci.
学ぶ。 略称はsci。
0.48
Technol. 1, 035007 (2020).
テクノル 1, 035007 (2020).
0.46
nications Physics 3, 1 (2020).
nications physics 3, 1 (2020)を参照。
0.83
[28] O. Danaci, S. Lohani, B. T. Kirby, and R. T. Glasser, Machine Learning: Science and Technology 2, 035014 (2021).
O. Danaci, S. Lohani, B. T. Kirby, R. T. Glasser, Machine Learning: Science and Technology 2, 035014 (2021)。
0.41
[46] S. Borah, B. Sarma, M. Kewming, G. J. Milburn, and
[46]s.ボラ、b.サルマ、m.キューミング、g.j.ミルバーン、そして
0.59
J. Twamley, Phys.
J. Twamley, Phys
0.40
Rev. Lett.
Rev. Lett.
0.43
127, 190403 (2021).
127, 190403 (2021).
0.42
[47] D. Gross, Y.
47] D. Gross, Y。
0.37
-K. Liu, S. T. Flammia, S. Becker,
-K。 Liu, S. T. Flammia, S. Becker
0.45
and [29] S. Ahmed, C. S. Mu˜noz, F. Nori, and A. F. Kockum,
そして [29]S.Ahmed,C.S.Mu snoz,F.Nori,A.F.Kock um
0.53
J. Eisert, Phys.
J. Eisert, Phys
0.40
Rev. Lett.
Rev. Lett.
0.43
105, 150401 (2010).
105, 150401 (2010).
0.42
Phys. Rev. Research 3, 033278 (2021).
Phys 第3巻033278号(2021年)。
0.36
[30] S. Lohani, T. A. Searles, B. T. Kirby, and R. Glasser,
S. Lohani, T. A. Searles, B. T. Kirby, R. Glasser.
0.41
IEEE Trans. Quantum Eng.
ieeeトランス。 量子化。
0.41
, 2103410 (2021).
, 2103410 (2021).
0.43
[31] S. Lohani, J. M. Lukens, D. E. Jones, T. A. Searles, R. T. Glasser, and B. T. Kirby, Phys.
S. Lohani, J. M. Lukens, D. E. Jones, T. A. Searles, R. T. Glasser, B. T. Kirby, Phys.
0.45
Rev. Research 3, 043145 (2021).
rev. research 3, 043145 (2021)。
0.35
[48] H. H¨affner, W. H¨ansel, C. F. Roos, J. Benhelm, D. Chekal kar, M. Chwalla, T. K¨orber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. G¨uhne, W. D¨ur, and R. Blatt, Nature 438, 643 (2005).
J. Benhelm, D. Chekal kar, M. Chwalla, T. K sorber, U.D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. G suhne, W. D sur, R. Blatt, Nature 438, 643 (2005)。 訳抜け防止モード: [48 ]H.H.saffner,W.H.sans el,C.F.Roos, J. Benhelm, D. Chekal kar, M. Chwalla, T. K sorber U.D. Rapol, M. Riebe, P. O. Schmidt, C. Becher O・G・ウーネ、W・D・ウール、R・ブラット、Nature 438。 643 ( 2005 ) .
0.74
[49] H. -J.
[49]H。 -j。
0.55
Sommers and K. Zyczkowski, J. Phys.
SommersとK. Zyczkowski、J. Phys。
0.45
A: Math. Gen. 36, 10083 (2003).
a: 数学。 第36代、第10083代(2003年)。
0.69
[32] S. Lohani, J. M. Lukens, R. T. Glasser, T. A. Searles,
[32]S. Lohani, J. M. Lukens, R. T. Glasser, T. A. Searles,
0.42
[50] K. Zyczkowski and H.
[50] K. Zyczkowski と H。
0.88
-J. Sommers, J. Phys.
-j。 Sommers、J. Phys。
0.59
A: Math. and B. T. Kirby, arXiv:2201.09134 (2022).
a: 数学。 B. T. Kirby, arXiv:2201.09134 (2022)。
0.75
Gen. 36, 10115 (2003).
第36代、第10115代(2003年)。
0.61
[33] G. Torlai, G. Mazzola, J. Carrasquilla, M. Troyer,
[34] G. Torlai, B. Timar, E. P. L. van Nieuwenburg, H. Levine, A. Omran, A. Keesling, H. Bernien, M. Greiner, V. Vuleti´c, M. D. Lukin, R. G. Melko, and M. Endres, Phys.
G. Torlai, B. Timar, E. P. L. van Nieuwenburg, H. Levine, A. Omran, A. Keesling, H. Bernien, M. Greiner, V. Vuleti ́c, M. D. Lukin, R. G. Melko, M. Endres, Phys。