qml.ClassicalShadow

class ClassicalShadow(bits, recipes, wire_map=None)

Bases: object

Class for classical shadow post-processing expectation values, approximate states, and entropies.

A ClassicalShadow is a classical description of a quantum state that is capable of reproducing expectation values of local Pauli observables, see 2002.08953.

The idea is to capture \(T\) local snapshots (given by the shots set in the device) of the state by performing measurements in random Pauli bases at each qubit. The measurement outcomes, denoted bits, as well as the choices of measurement bases, recipes, are recorded in two (T, len(wires)) integer tensors, respectively.

From the \(t\)-th measurement, we can reconstruct the local_snapshots (see methods)

\[\rho^{(t)} = \bigotimes_{i=1}^{n} 3 U^\dagger_i |b_i \rangle \langle b_i | U_i - \mathbb{I},\]

where \(U_i\) is the rotation corresponding to the measurement (e.g. \(U_i=H\) for measurement in \(X\)) of qubit \(i\) at snapshot \(t\) and \(|b_i\rangle = (1 - b_i, b_i)\) the corresponding computational basis state given the output bit \(b_i\).

From these local snapshots, one can compute expectation values of local Pauli strings, where locality refers to the number of non-Identity operators. The accuracy of the procedure is determined by the number of measurements \(T\) (shots). To target an error \(\epsilon\), one needs of order \(T = \mathcal{O}\left( \log(M) 4^\ell/\epsilon^2 \right)\) measurements to determine \(M\) different, \(\ell\)-local observables.

One can in principle also reconstruct the global state \(\sum_t \rho^{(t)}/T\), though it is not advisable nor practical for larger systems due to its exponential scaling.

Note

As per arXiv:2103.07510, when computing multiple expectation values it is advisable to directly estimate the desired observables by simultaneously measuring qubit-wise-commuting terms. One way of doing this in PennyLane is via Hamiltonian and setting grouping_type="qwc". For more details on this topic, see our demo on estimating expectation values with classical shadows.

Parameters

• bits (tensor) – recorded measurement outcomes in random Pauli bases.

• recipes (tensor) – recorded measurement bases.

• wire_map (list[int]) – list of the measured wires in the order that they appear in the columns of bits and recipes. If None, defaults to range(n), where n is the number of measured wires.

See also

Demo on Estimating observables with classical shadows in the Pauli basis, classical_shadow()

Example

We obtain the bits and recipes via classical_shadow() measurement:

dev = qml.device("default.qubit", wires=range(2), shots=1000)
@qml.qnode(dev)
def qnode(x):
    qml.Hadamard(0)
    qml.CNOT((0,1))
    qml.RX(x, wires=0)
    return qml.classical_shadow(wires=range(2))

bits, recipes = qnode(0)
shadow = qml.ClassicalShadow(bits, recipes)

After recording these T=1000 quantum measurements, we can post-process the results to arbitrary local expectation values of Pauli strings. For example, we can compute the expectation value of a Pauli string

>>> shadow.expval(qml.X(0) @ qml.X(1), k=1)
array(0.972)

or of a Hamiltonian:

>>> H = qml.Hamiltonian([1., 1.], [qml.Z(0) @ qml.Z(1), qml.X(0) @ qml.X(1)])
>>> shadow.expval(H, k=1)
array(1.917)

The parameter k is used to estimate the expectation values via the median of means algorithm (see 2002.08953). The case k=1 corresponds to simply taking the mean value over all local snapshots. k>1 corresponds to splitting the T local snapshots into k equal parts, and taking the median of their individual means. For the case of measuring only in the Pauli basis, there is no advantage expected from setting k>1.

Attributes

snapshots

The number of snapshots in the classical shadow measurement.

snapshots

The number of snapshots in the classical shadow measurement.

Methods

entropy(wires[, snapshots, alpha, k, base])	Compute entropies from classical shadow measurements.
expval(H[, k])	Compute expectation value of an observable \(H\).
global_snapshots([wires, snapshots])	Compute the T x 2**n x 2**n global snapshots
local_snapshots([wires, snapshots])	Compute the T x n x 2 x 2 local snapshots

entropy(wires, snapshots=None, alpha=2, k=1, base=None)

Compute entropies from classical shadow measurements.

Compute general Renyi entropies of order \(\alpha\) for a reduced density matrix \(\rho\) in terms of

\[S_\alpha(\rho) = \frac{1}{1-\alpha} \log\left(\text{tr}\left[\rho^\alpha \right] \right).\]

There are two interesting special cases: In the limit \(\alpha \rightarrow 1\), we find the von Neumann entropy

\[S_{\alpha=1}(\rho) = -\text{tr}(\rho \log(\rho)).\]

In the case of \(\alpha = 2\), the Renyi entropy becomes the logarithm of the purity of the reduced state

\[S_{\alpha=2}(\rho) = - \log\left(\text{tr}(\rho^2) \right).\]

Since density matrices reconstructed from classical shadows can have negative eigenvalues, we use the algorithm described in 1106.5458 to project the estimator to the closest valid state.

Warning

Entropies are non-linear functions of the quantum state. Accuracy bounds on entropies with classical shadows are not known exactly, but scale exponentially in the subsystem size. It is advisable to only compute entropies for small subsystems of a few qubits. Further, entropies as post-processed by this class method are currently not automatically differentiable.

Parameters

• wires (Iterable[int]) – The wires over which to compute the entropy of their reduced state. Note that the computation scales exponentially in the number of wires for the reduced state.

• snapshots (Iterable[int] or int) – Only compute a subset of local snapshots. For snapshots=None (default), all local snapshots are taken. In case of an integer, a random subset of that size is taken. The subset can also be explicitly fixed by passing an Iterable with the corresponding indices.

• alpha (float) – order of the Renyi-entropy. Defaults to alpha=2, which corresponds to the purity of the reduced state. This case is straight forward to compute. All other cases alpha!=2 necessitate computing the eigenvalues of the reduced state and thus may lead to longer computations times. Another special case is alpha=1, which corresponds to the von Neumann entropy.

• k (int) – Allow to split the snapshots into k equal parts and estimate the snapshots in a median of means fashion. There is no known advantage to do this for entropies. Thus, k=1 is default and advised.

• base (float) – Base to the logarithm used for the entropies.

Returns

Entropy of the chosen subsystem.

Return type

float

Example

For the maximally entangled state of n qubits, the reduced state has two constant eigenvalues \(\frac{1}{2}\). For constant distributions, all Renyi entropies are equivalent:

wires = 4
dev = qml.device("default.qubit", wires=range(wires), shots=1000)

@qml.qnode(dev)
def max_entangled_circuit():
    qml.Hadamard(wires=0)
    for i in range(1, wires):
        qml.CNOT(wires=[0, i])
    return qml.classical_shadow(wires=range(wires))

bits, recipes = max_entangled_circuit()
shadow = qml.ClassicalShadow(bits, recipes)

entropies = [shadow.entropy(wires=[0], alpha=alpha) for alpha in [1., 2., 3.]]

>>> np.isclose(entropies, entropies[0], atol=1e-2)
[ True,  True,  True]

For non-uniform reduced states that is not the case anymore and the entropy differs for each order alpha:

@qml.qnode(dev)
def qnode(x):
    for i in range(wires):
        qml.RY(x[i], wires=i)

    for i in range(wires - 1):
        qml.CNOT((i, i + 1))

    return qml.classical_shadow(wires=range(wires))

x = np.linspace(0.5, 1.5, num=wires)
bitstrings, recipes = qnode(x)
shadow = qml.ClassicalShadow(bitstrings, recipes)

>>> [shadow.entropy(wires=wires, alpha=alpha) for alpha in [1., 2., 3.]]
[1.5419292874423107, 1.1537924276625828, 0.9593638767763727]

expval(H, k=1)

Compute expectation value of an observable \(H\).

The canonical way of computing expectation values is to simply average the expectation values for each local snapshot, \(\langle O \rangle = \sum_t \text{tr}(\rho^{(t)}O) / T\). This corresponds to the case k=1. In the original work, 2002.08953, it has been proposed to split the T measurements into k equal parts to compute the median of means. For the case of Pauli measurements and Pauli observables, there is no advantage expected from setting k>1.

One of the main perks of classical shadows is being able to compute many different expectation values by classically post-processing the same measurements. This is helpful in general as it may help save quantum circuit executions.

Parameters

• H (qml.Observable) – Observable to compute the expectation value

• k (int) – Number of equal parts to split the shadow’s measurements to compute the median of means. k=1 (default) corresponds to simply taking the mean over all measurements.

Returns

expectation value estimate.

Return type

float

Example

dev = qml.device("default.qubit", wires=range(2), shots=1000)
@qml.qnode(dev)
def qnode(x):
    qml.Hadamard(0)
    qml.CNOT((0,1))
    qml.RX(x, wires=0)
    return qml.classical_shadow(wires=range(2))

bits, recipes = qnode(0)
shadow = qml.ClassicalShadow(bits, recipes)

Compute Pauli string observables

>>> shadow.expval(qml.X(0) @ qml.X(1), k=1)
array(1.116)

or of a Hamiltonian using the same measurement results

>>> H = qml.Hamiltonian([1., 1.], [qml.Z(0) @ qml.Z(1), qml.X(0) @ qml.X(1)])
>>> shadow.expval(H, k=1)
array(1.9980000000000002)

global_snapshots(wires=None, snapshots=None)

Compute the T x 2**n x 2**n global snapshots

Warning

Classical shadows are not intended to reconstruct global quantum states. This method requires exponential scaling of measurements for accurate representations. Further, the output scales exponentially in the output dimension, and is therefore not practical for larger systems. A warning is raised for systems of sizes n>16.

Parameters

• wires (Iterable[int]) – The wires over which to compute the snapshots. For wires=None (default) all n qubits are used.

• snapshots (Iterable[int] or int) – Only compute a subset of local snapshots. For snapshots=None (default), all local snapshots are taken. In case of an integer, a random subset of that size is taken. The subset can also be explicitly fixed by passing an Iterable with the corresponding indices.

Returns

The global snapshots tensor of shape (T, 2**n, 2**n) containing the density matrices for each snapshot measurement.

Return type

tensor

Example

We can approximately reconstruct a Bell state:

dev = qml.device("default.qubit", wires=range(2), shots=1000)
@qml.qnode(dev)
def qnode():
    qml.Hadamard(0)
    qml.CNOT((0,1))
    return classical_shadow(wires=range(2))

bits, recipes = qnode()
shadow = ClassicalShadow(bits, recipes)
shadow_state = np.mean(shadow.global_snapshots(), axis=0)

bell_state = np.array([[0.5, 0, 0, 0.5], [0, 0, 0, 0], [0, 0, 0, 0], [0.5, 0, 0, 0.5]])

>>> np.allclose(bell_state, shadow_state, atol=1e-1)
True

local_snapshots(wires=None, snapshots=None)

Compute the T x n x 2 x 2 local snapshots

For each qubit and each snapshot, compute \(3 U_i^\dagger |b_i \rangle \langle b_i| U_i - 1\)

Parameters

• wires (Iterable[int]) – The wires over which to compute the snapshots. For wires=None (default) all n qubits are used.

• snapshots (Iterable[int] or int) – Only compute a subset of local snapshots. For snapshots=None (default), all local snapshots are taken. In case of an integer, a random subset of that size is taken. The subset can also be explicitly fixed by passing an Iterable with the corresponding indices.

Returns

The local snapshots tensor of shape (T, n, 2, 2) containing the local local density matrices for each snapshot and each qubit.

Return type

tensor

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