Quantum Machine Learning Tutorial

A Hands-on Tutorial for Machine Learning Practitioners and Researchers

Chapter 2.3 Quantum Read-in and Read-out protocols

The terms quantum read-in and read-out refer to the processes of transferring information between classical systems and quantum systems. These are fundamental steps in the workflow of quantum machine learning, responsible for loading data and extracting results.

Quantum read-in and read-out pose significant bottlenecks in leveraging quantum computing to address classical computational tasks. As emphasized in [@aaronson2015read], while quantum algorithms can offer exponential speed-ups in specific problem domains, these advantages can be negated if the processes of loading classical data into quantum systems (read-in) or extracting results from quantum systems (read-out) are inefficient. Specifically, the high-dimensional nature of quantum states and the constraints on measurement precision often lead to overheads that scale poorly with problem size. These challenges underscore the importance of optimizing quantum read-in and read-out protocols to realize the full potential of quantum computing. Below is a detailed introduction to quantum read-int and read-out protocols, including the basic concept and several typical algorithms.

Quantum read-in protocols

Quantum read-in refers to the process of encoding classical information into quantum systems that can be manipulated by a quantum computer, which can be regarded as the classical-to-quantum mapping. It acts as a bridge to utilize quantum algorithms to solve classical problems in quantum computing. Here, we will introduce several typical encoding methods, including basis encoding, amplitude encoding, angle encoding, and quantum random access memory.

Basis encoding

Basis encoding is a basic method for processing classical data that can be represented in binary form. Given a classical binary vector $\bm{x} \in \{0, 1\}^N$, this encoding technique maps the vector directly into a quantum computational basis state as follows:

$$\ket{\psi} = \ket{\bm{x}_1,…,\bm{x}_N}.$$

In this process, $N$ qubits are required to represent a binary vector of length $N$. To prepare the corresponding quantum state $\ket{\psi}$, an $X$ gate is applied to each qubit where the corresponding bit value is 1. The overall quantum state preparation can be expressed as: $$\ket{\psi} = \bigotimes_{i=1}^{N} X^{\bm{x}_i} \ket{0}^{\otimes N},$$ where $\ket{0}^{\otimes N}$ represents an initial state of all qubits set to $\ket{0}$, and $X^{\bm{x}_i}$ means applying the $X$ gate only if $\bm{x}_i = 1$.

Example of Basis encoding. Consider encoding the integer $6$, which has the binary representation $\bm{x} = (1, 1, 0)$. The corresponding quantum state is $\ket{110}$. This state can be implemented by applying $X$ gates to the first and second qubits.

Amplitude encoding

Amplitude encoding is a technique that maps classical data into the amplitudes of a quantum state. Given a vector $\bm{x} \in \mathbb{C}^{2^N}$ containing complex values, we first apply $L_2$ normalization to obtain a normalized vector $$ \hat{\bm{x}} = \frac{\bm{x}}{|\bm{x}|_2},$$

where $|\bm{x}|_2$ is the Euclidean norm. This ensures that the normalized vector $\hat{\bm{x}}$ satisfies

$$\sum_{i=0}^{2^N-1} |\hat{\bm{x}}_i|^2 = 1$$.

The corresponding quantum state is then expressed as $$\ket{\psi} = \sum_{i=0}^{2^N-1} \hat{\bm{x}}_i \ket{i},$$ where $\ket{i}$ representing the $N$-qubit computational basis states.

Example of Amplitude encoding. Consider encoding a normalized vector $\bm{x} = (\bm{x}_0, \bm{x}_1) \in \mathbb{C}^2$ into the quantum state $\ket{\psi} = \bm{x}_0 \ket{0} + \bm{x}_1 \ket{1}$. This can be achieved by applying a rotation gate $U=R_Y(\theta)$ to the initial state $\ket{0}$, where $\theta = 2 \arccos(\bm{x}_0)$.

Amplitude encoding is highly efficient because it allows an exponentially large vector of length $2^N$ to be represented using only $N$ qubits. However, preparing this quantum state requires constructing a unitary transformation $U$ such that $\ket{\psi} = U \ket{0}^{\otimes N}$. Efficiently finding such transformations can be challenging and is an active area of research.

Angle encoding

Basis encoding and amplitude encoding are fundamental techniques for mapping classical data to quantum states, but each comes with distinct resource costs. Basis encoding requires a number of qubits equal to the dimensionality of the binary representation of classical data and necessitates minimal gate operations for state preparation. In contrast, amplitude encoding is highly compact in terms of qubits, using only the logarithmic of the data dimensionality, but it involves a significant gate complexity.

To address this limitation, an alternative is angle encoding. The core idea of angle encoding is to embed classical data into a quantum state through rotation angles.

Given a real-valued vector $\bm{x} \in \mathbb{R}^{N}$, the encoded quantum state can be represented as:

$$\ket{\psi} = \bigotimes_{i=0}^{N-1} R_{\sigma}(\bm{x}_i) \ket{0}^{\otimes N},$$

where $\sigma \in \{X, Y, Z\}$ denotes a Pauli operator. Since Pauli rotation gates are $2\pi$-periodic, it is essential to scale each element $\bm{x}_i$ into the range $[0, \pi)$ to ensure that different values are encoded into distinct quantum states.

A key advantage of angle encoding is its ability to introduce nonlinearity. By mapping classical data into the parameters of quantum rotation gates, angle encoding leverages trigonometric functions to naturally capture non-linear relationships. This property is particularly important in quantum machine learning, where nonlinearity is essential for models to learn complex patterns in data, such as non-linearly separable decision boundaries.

Quantum Random Access Memory (QRAM)

Basis encoding, amplitude encoding, and angle encoding are generally designed to encode a single item of data at one time, which makes it challenging to process complicated classical datasets. The QRAM [@giovannetti2008quantum], analogous to classical RAM, aims to simultaneously store, address, and access multiple quantum states.

QRAM consists of two types of qubits: data qubits for storing classical data and address qubits for addressing. Given a classical dataset $\mathcal{D}={\bm{x}^{(j)}}_{j=0}^{M-1}$ with $M$ training examples, assume we separately encode each data item into a quantum state $\ket{\bm{x}^{(j)}}_d$ using one of the encoding methods above. The QRAM can be constructed as follows: (1) Prepare an $N_a$-qubit address register where $N_a=\lceil \log_2(M) \rceil$; (2) Associate each data state $\ket{\bm{x}^{(j)}}_d$ with corresponding address state $\ket{j}_a$. The whole dataset is therefore encoded into a quantum state of the form

$$\ket{\mathcal{D}}=\sum_{j=0}^{M-1}\frac{1}{\sqrt{M}}\ket{j}_a\ket{\bm{x}^{(j)}}_d.$$

The subscript $d$ in $\ket{\bm{x}^{(j)}}_d$ indicates that this quantum state resides in the data register, differentiating it from address qubits, which are denoted with the subscript $a$ (e.g., $\ket{j}_a$). This convention helps to distinguish between the roles of data and address qubits in QRAM operations.

Example of QRAM Encoding. Consider a dataset $\mathcal{D} = {2, 3}$. Using basis encoding, each sample is first converted into a two-qubit quantum state: ${\ket{10}_d, \ket{11}_d}$. Each data state is then assigned an address state, $\ket{0}_a$ for the first state $\ket{10}_d$ and $\ket{1}_a$ for the second state $\ket{11}_d$. The resulting QRAM-encoded state is: $$\ket{\mathcal{D}} = \frac{1}{\sqrt{2}} \left( \ket{0}_a \ket{10}_d + \ket{1}_a \ket{11}_d \right).$$

QRAM allows the dataset $\mathcal{D}$ to be stored in a coherent quantum superposition, enabling simultaneous access to all data items through the entanglement of address and data qubits. While QRAM is theoretically powerful, its practical implementation remains a significant challenge due to the need for a large number of qubits and quantum operations.

Quantum read-out protocols

Quantum read-out refers to the process of translating the quantum state resulting from a quantum computation into classical data, enabling further processing, interpretation, or optimization in classical systems. This process can be considered the inverse operation of quantum read-in, representing a quantum-to-classical mapping.

Based on the completeness of the information obtained during the read-out process, quantum read-out protocols can be broadly categorized into two types, i.e., full information and partial information read-out protocols. These protocols enable tailored read-out processes that match the requirements of different quantum applications, ranging from tomography to optimization and machine learning tasks.

Full information read-out protocol

The full information read-out protocol is used to completely reconstruct the quantum state, which is used to fully understand the quantum system’s behavior. The most general approach to implementing this protocol is through quantum state tomography (QST).

QST involves performing quantum measurements, gathering measurement statistics, and using classical post-processing to reconstruct the quantum state. In what follows, we introduce two reconstruction techniques broadly used in QST, i.e., linear inversion and maximum likelihood estimation (MLE).

QST with linear inversion. Linear inversion is a straightforward method to reconstruct the quantum state from measurement data by directly solving linear systems of equations. Let $\rho$ be the explored quantum state and ${E_i}$ be a set of measurements. According to the Born rule, the probability of measurement outcome $i$ is given by $$P(E_i|\rho) = \text{Tr}(\rho E_i).$$ In practice, $P(E_i|\rho)$ is not directly accessible but is approximated by the frequency $p_i$ of measurement outcome $i$ over multiple measurements. By the law of large numbers, as the number of measurements increases, $p_i$ converges to the true probability $P(E_i|\rho)$. Collecting measurements across all bases, we obtain a linear system $$\begin{bmatrix} \text{Tr}(\rho E_0) \ \text{Tr}(\rho E_1) \ \vdots \end{bmatrix} = \begin{bmatrix} \vec{E}_0^\dagger \cdot \vec{\rho} \ \vec{E}_1^\dagger \cdot \vec{\rho} \ \vdots \end{bmatrix} = A\vec{\rho} \approx \bm{p} = \begin{bmatrix} p_0 \ p_1 \ \vdots \end{bmatrix},$$ where $\vec{E}$ and $\vec{\rho}$ refer to the vector representations of matrices $E_i$ and $\rho$, respectively. The vector representation of a matrix is obtained by stacking its columns into a single-column vector. For example, the vector representation of a $2\times 2$ identity matrix is $\vec{\mathbb{I}}_2=[1,0,0,1]^T$. The matrix $A$ is constructed such that each row corresponds to the vector representation of the measurement operator, i.e., $A=[\vec{E}_0^\dagger;\vec{E}_1^\dagger;…]$. The vector $\bm{p}$ contains the measured frequencies $p_i$.

Assuming the measurements are tomographically complete, i.e., ${E_i}$ forms a basis for the system’s Hilbert space, we can reconstruct $\rho$ by solving the following linear systems of equations, i.e., $$\vec{\rho} = (A^T A)^{-1} A^T \bm{p}.$$

A common strategy is to use Pauli operators as measurement bases ${E_i}$. The density matrix $\rho$ of an $N$-qubit system can be expanded in terms of the Pauli basis as: $$\rho = \frac{1}{2^N} \sum_{i=0}^{4^N - 1} c_i P_i, \quad c_i \in \mathbb{R}, \quad P_i \in {I, X, Y, Z}^{\otimes N},$$ where the coefficients $c_i$ represent projections of $\rho$ onto the Pauli basis, calculated as: $$c_i = \text{Tr}(\rho P_i).$$ To fully reconstruct $\rho$, the quantum state must theoretically be measured in all $4^N - 1$ Pauli bases to estimate each $c_i$.

::: tcolorbox The Pauli basis consists of four Hermitian matrics $I$, $X$, $Y$ and $Z$ introduced in Figure 1.1{reference-type=“ref” reference=“tab:Q-gates”}. These operators form a complete basis for the space of $2\times 2$ complex matrices. For $N$-qubit systems, the tensor products of these single-qubit operators span the space of $2^N \times 2^N$ complex matrics. This makes the Pauli basis essential for representing quantum states, observables, and their transformations. :::

A key limitation of linear inversion is that it does not guarantee a valid density matrix, as the estimated quantum state may lack properties such as positive semi-definiteness, especially with limited measurements.

Maximum Likelihood Estimation (MLE). To ensure physical constraints on the quantum state during reconstruction, MLE is introduced. MLE reconstructs $\rho$ by maximizing the likelihood of observing the measurement outcomes, subject to the constraints that $\rho$ is Hermitian, positive semi-definite, and trace one. The likelihood function is given by $$L(\rho) = \prod_i \text{Tr}(\rho E_i)^{p_i}.$$ Reconstructing $\rho$ then reduces to solving the following optimization problem $$\argmax_{\rho’} L(\rho’), \quad \text{s.t.} \quad \rho’ \succeq 0, \quad \rho’ = \rho’^\dagger, \quad \text{Tr}(\rho’) = 1.$$ Solving this typically requires iterative numerical optimization, which can be computationally intensive.

A common challenge across all quantum state tomography (QST) methods, including linear inversion and MLE, is the exponential computational cost with respect to the number of qubits. Specifically, the number of parameters to reconstruct grows exponentially with the system size, making QST methods feasible only for small-qubit systems in practice. This limitation underscores the need for scalable approaches to quantum state characterization in larger quantum systems.

Partial information read-out protocol

The partial information read-out protocol focuses on extracting specific, task-relevant information from a quantum state without reconstructing the entire density matrix. This protocol is efficient and can be applied to comprehend large-qubit systems. According to the type of collected information, current partial read-out protocols can be categorized into three classes, i.e., sampling, expectation value estimation, and shadow tomography.

Sampling. Sampling involves repeatedly measuring the quantum state in the computational basis to estimate the probability distribution over bit-strings. Given a state $\ket{\psi}$, the probability of observing a specific computational basis $\ket{i}$ is given by $$P(i) = \left| \braket{\psi|i}\right|^2.$$ The empirical frequency of each outcome from repeated measurements provides an estimate of $P(i)$. Sampling is particularly useful in the following applications

  • Sampling over complicated distributions. Quantum states can represent complex probability distributions that are difficult to sample classically. Quantum sampling allows efficient exploration of these distributions for specific applications, such as probabilistic modeling and Markov chain Monte Carlo.

  • Optimization problems. Sampling can identify high-probability bitstrings in quantum algorithms, such as the Quantum Approximate Optimization Algorithm [@farhi2014quantum] and Grover search [@grover1996fast], where these bitstrings often correspond to optimal or near-optimal solutions.

  • Verification. Sampling facilitates the comparison of a quantum circuit’s output with theoretical expectations or desired distributions, helping to verify the quantum systems.

Expectation value estimation. For a wide class of quantum computation problems, such as quantum chemistry and quantum many-body physics, the computation outcome refers to the estimation of the expectation values of certain observables on the evolved quantum state.

An observable $O\in \mathbb{C}^{2^N \times 2^N}$ mentioned here is a Hermitian operator that represents a measurable physical quantity. For an $N$-qubit system, $O$ can be expressed in terms of a Pauli basis expansion, i.e., $$O = \sum_{i=1}^{4^N} \alpha_i P_i, \quad P_i \in \{I, X, Y, Z\}^{\otimes N}, \quad \alpha_i \in \mathbb{R}.$$ where $P_i$ is the tensor product of Pauli operators.

The expectation value of an observale $O$ with respect to an $N$-qubit state $\rho$ is $$ \braket{O} = \text{Tr}(\rho O).$$

Substituting the Pauli expansion of $O$, the expectation value is expressed as the weighted sum of the expectation values of each Pauli basis term due to the linearity of the trace operation, i.e., $$\braket{O} = \sum_{i=1}^{4^N} \alpha_i \text{Tr}(\rho P_i) = \sum_{i=1}^{4^N} \alpha_i \braket{P_i}.$$

To estimate the expectation value of each individual Pauli term $P_i$, the quantum state $\rho$ must be measured in the basis of the eigenstates of $P_i$. The measurement outcome is then associated with the corresponding eigenvalue of $P_i$. Fortunately, the eigenstates and eigenvalues of $P_i$ can be derived from the eigenstates and eigenvalues of its constituent single-qubit Pauli operators $P_{ij}$. Specifically:

  • Eigenvalues: The eigenvalues of $P_i$ are the product of the eigenvalues of each single-qubit Pauli operator $P_{ij}$. For example, if the eigenvalues of $P_{ij}$ are $\pm 1$, then the eigenvalues of $P_i$ are products of these individual eigenvalues and remain in ${ \pm 1 }$.

  • Eigenstates: The eigenstates of $P_i$ are the tensor products of the eigenstates of the single-qubit Pauli operators $P_{ij}$. If $\ket{\lambda_{ijk}}$ is one of the eigenstate of $P_{ij}$, then one eigenstate of $P_i$ is $\bigotimes_{j=1}^N \ket{\lambda_{ijk}}$.

This structure allows $P_i$ to be analyzed in terms of its simpler single-qubit components, significantly simplifying the process of determining the measurement basis for expectation value estimation. By repeating the measurements $M$ times and obtaining the corresponding measurement results ${r_j}_{j=1}^{M}$, the statistical value of $\braket{P_i}$ can be estimated by

$$\braket{\hat{P}i} = \frac{1}{M} \sum{j=1}^{M} r_j.$$ The expectation value of the observable $O$ is therefore statistically estimated by $\braket{\hat{O}}=\sum_{i=0}^{K-1}\alpha_i \braket{\hat{P}_i}$.

A key step in the process is to measure the quantum system in the basis of the eigenstates of $P_i$. If $P_i$ is diagonal in the computational basis (e.g., a tensor product of Pauli-Z operators), we can directly measure the state without additional operations. Otherwise (e.g., for Pauli-X or Pauli-Y operators), we need to apply a unitary transformation to rotate the quantum state into the desired basis. Specifically, when measuring in the Pauli-X basis (i.e., $\ket{+}$ and $\ket{-}$), a Hadamard gate $H$ is applied to the state $\rho$, i.e., $$\rho’ = H \rho H.$$

When measuring in the Pauli-Y basis (i.e., $\frac{\ket{0} + i\ket{1}}{\sqrt{2}}$ and $\frac{\ket{0} - i\ket{1}}{\sqrt{2}}$), a phase gate $S = \sqrt{Z}$ followed by a Hadamard gate $H$ is applied, i.e., $$\rho’ = S^\dagger H \rho H S.$$ Measuring the state $\rho’$ in the computational basis is equivalent to measuring the state $\rho$ in the corresponding Pauli basis.

Shadow tomography. Performing full QST requires an exponential number of copies of the quantum state, making it impractical for systems beyond a small number of qubits. Instead of reconstructing the complete density matrix, shadow tomography focuses on efficiently obtaining specific properties of a quantum state, such as the expectation values of many observables.

Definition. Given an unknown $D$-dimensional quantum state $\rho$, as well as $M$ observables $O_1,…,O_M$, output real numbers $b_1,…,b_M$ such that $\left|b_i-\text{Tr}(O_i\rho)\right|\leq\epsilon$ for all $i$, with success probability at least $1-\delta$. Do this via a measurement of $\rho^{\otimes k}$, where $k=k(D,M,\epsilon,\delta)$ is as small as possible.

@aaronson2018shadow proved that the shadow tomography problem can be solved using a polylogarithmic number of copies of states in terms of the dimension $D$ and number $M$ of observables. This result demonstrates that it is possible to estimate the expectation values of exponentially many observables for a quantum state of exponential dimension using only a polynomial number of measurements.

The central idea of shadow tomography is to create a compact measurement classical representation, or “shadow,” of a quantum state that encodes sufficient information to estimate many properties of the state. Building on this concept, @huang2020predicting proposed a more practical and efficient approach, termed classical shadow, which uses randomized measurements to construct this classical representation. The classical shadow approach consists of the following steps:

  1. Randomized measurements: Perform random unitary transformations on the quantum state and measure the transformed state in the computational basis. These random transformations can be drawn from specific ensembles, such as Clifford gates or local random rotations, which ensure that the measurement outcomes capture the essential properties of the quantum state.

  2. Classical shadow construction: Using the measurement results, construct a classical shadow of the quantum state. This compact representation encodes the quantum state in a way that allows for the efficient estimation of properties.

  3. Property estimation: Use the classical shadow to compute the desired properties of the quantum state, such as expectation values of specific observables, subsystem entropies, or fidelities with known states.

Shadow tomography requires exponentially fewer measurements compared to full quantum state tomography, making it a practical solution for large-scale quantum systems. Moreover, the shadow of a quantum state serves as a versatile representation, enabling the efficient estimation of various properties such as expectation values, entanglement measures, and subsystem correlations.