How Do We Mark the Good Stuff?

The goal of amplitude amplification is to increase the probability of favorable solutions. In our example, we use the amplitude amplification technique to increase the probability of the winning shell.

Amplitude Amplification

The Strange Art of Amplifying Success

October 22, 2025•5 min

Eight shells, one hidden gem, and a quantum trick that beats pure chance. Learn how quantum amplitude amplification uses simple geometry to bend probability.

In the previous post we learned that this is a two-step procedure. First, the OracleAn oracle is a black-box function that encodes information about a problem—typically deciding whether a given input satisfies some condition. It’s implemented as a quantum operation that can be queried in superposition, allowing a quantum algorithm to extract global properties of the function more efficiently than classical methods. Oracles are central to algorithms like Grover’s and Deutsch–Jozsa, where they guide the computation without revealing internal details.
Learn more about Oracle reverses the sign of the AmplitudeIn quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
Learn more about Amplitude that corresponds to the Basis StateA basis state in quantum computing is one of the fundamental states that form the building blocks of a quantum system’s state space. For a single qubit, the basis states are and ; any other qubit state is a superposition of these. In systems with multiple qubits, basis states are all possible combinations of s and s (e.g., , , , and ), forming an orthonormal basis for the system’s Hilbert space.
Learn more about Basis State that represents the correct answer. Then the diffusion stepThe diffusion operator (also called the Grover diffusion operator) is a unitary operation that inverts the amplitude of each state about the average amplitude. It’s used in Grover’s search algorithm to amplify the probability of the marked (target) state. Mathematically, it’s represented as , where is the uniform superposition state.
Learn more about Diffusion Operator reflects all AmplitudesIn quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
Learn more about Amplitude by the mean, boosting the one that is the farthest from the mean. That is the marked Quantum StateA quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State. The one with a negative Amplitude.In quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
Learn more about Amplitude

So, the Oracle'sAn oracle is a black-box function that encodes information about a problem—typically deciding whether a given input satisfies some condition. It’s implemented as a quantum operation that can be queried in superposition, allowing a quantum algorithm to extract global properties of the function more efficiently than classical methods. Oracles are central to algorithms like Grover’s and Deutsch–Jozsa, where they guide the computation without revealing internal details.
Learn more about Oracle only job is to reverse the sign of the AmplitudeIn quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
Learn more about Amplitude corresponding to the correct answer, as ? shows.

The oracle marks the correct basis state by reversing its phase, not its probability

But what does it actually mean to reverse the AmplitudeIn quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
Learn more about Amplitude of a Quantum State?A quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State

After all, quantum superpositionSuperposition in quantum computing means a quantum bit (qubit) can exist in multiple states (0 and 1) at the same time, rather than being limited to one like a classical bit. Mathematically, it’s a linear combination of basis states with complex probability amplitudes. This allows quantum computers to process many possible inputs simultaneously, enabling exponential speedups for certain problems.
Learn more about Superposition tells us that every Quantum SystemA quantum system is any physical system that is subject to the laws of quantum mechanics, whereby quantities such as energy or spin can only assume discrete (quantized) values. Its behavior is described by a wave function that encodes the probabilities of possible measurement results.
Learn more about Quantum System is in a complexA complex number is a number that has two parts: a real part and an imaginary part, written as , where . The real part behaves like ordinary numbers, while the imaginary part represents a direction perpendicular to the real axis on the complex plane. Complex numbers let us represent and calculate quantities involving square roots of negative numbers.
Learn more about Complex Number linear combination of its Basis States.A basis state in quantum computing is one of the fundamental states that form the building blocks of a quantum system’s state space. For a single qubit, the basis states are and ; any other qubit state is a superposition of these. In systems with multiple qubits, basis states are all possible combinations of s and s (e.g., , , , and ), forming an orthonormal basis for the system’s Hilbert space.
Learn more about Basis State

• A single Quantum BitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
  Learn more about Quantum Bit has two Basis StatesA basis state in quantum computing is one of the fundamental states that form the building blocks of a quantum system’s state space. For a single qubit, the basis states are and ; any other qubit state is a superposition of these. In systems with multiple qubits, basis states are all possible combinations of s and s (e.g., , , , and ), forming an orthonormal basis for the system’s Hilbert space.
  Learn more about Basis State: is a basis state.
  Learn more about and is a basis state.
  Learn more about .
• A two-Quantum BitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
  Learn more about Quantum Bit Quantum SystemA quantum system is any physical system that is subject to the laws of quantum mechanics, whereby quantities such as energy or spin can only assume discrete (quantized) values. Its behavior is described by a wave function that encodes the probabilities of possible measurement results.
  Learn more about Quantum System has four Basis StatesA basis state in quantum computing is one of the fundamental states that form the building blocks of a quantum system’s state space. For a single qubit, the basis states are and ; any other qubit state is a superposition of these. In systems with multiple qubits, basis states are all possible combinations of s and s (e.g., , , , and ), forming an orthonormal basis for the system’s Hilbert space.
  Learn more about Basis State: , , , and .
• A three-Quantum BitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
  Learn more about Quantum Bit Quantum SystemA quantum system is any physical system that is subject to the laws of quantum mechanics, whereby quantities such as energy or spin can only assume discrete (quantized) values. Its behavior is described by a wave function that encodes the probabilities of possible measurement results.
  Learn more about Quantum System has eight Basis StatesA basis state in quantum computing is one of the fundamental states that form the building blocks of a quantum system’s state space. For a single qubit, the basis states are and ; any other qubit state is a superposition of these. In systems with multiple qubits, basis states are all possible combinations of s and s (e.g., , , , and ), forming an orthonormal basis for the system’s Hilbert space.
  Learn more about Basis State. And so on.

Each of these Basis StatesA basis state in quantum computing is one of the fundamental states that form the building blocks of a quantum system’s state space. For a single qubit, the basis states are and ; any other qubit state is a superposition of these. In systems with multiple qubits, basis states are all possible combinations of s and s (e.g., , , , and ), forming an orthonormal basis for the system’s Hilbert space.
Learn more about Basis State has a complexA complex number is a number that has two parts: a real part and an imaginary part, written as , where . The real part behaves like ordinary numbers, while the imaginary part represents a direction perpendicular to the real axis on the complex plane. Complex numbers let us represent and calculate quantities involving square roots of negative numbers.
Learn more about Complex Number AmplitudeIn quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
Learn more about Amplitude that denotes its contribution to the overall Quantum System.A quantum system is any physical system that is subject to the laws of quantum mechanics, whereby quantities such as energy or spin can only assume discrete (quantized) values. Its behavior is described by a wave function that encodes the probabilities of possible measurement results.
Learn more about Quantum System ComplexA complex number is a number that has two parts: a real part and an imaginary part, written as , where . The real part behaves like ordinary numbers, while the imaginary part represents a direction perpendicular to the real axis on the complex plane. Complex numbers let us represent and calculate quantities involving square roots of negative numbers.
Learn more about Complex Number here means the AmplitudeIn quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
Learn more about Amplitude is a two-dimensional ComplexA complex number is a number that has two parts: a real part and an imaginary part, written as , where . The real part behaves like ordinary numbers, while the imaginary part represents a direction perpendicular to the real axis on the complex plane. Complex numbers let us represent and calculate quantities involving square roots of negative numbers.
Learn more about Complex Number. This is similar to a 2D VectorA vector is a mathematical object that has both magnitude (size) and direction. It’s often represented as an arrow or as an ordered list of numbers (components) that describe its position in space, such as . Vectors are used to represent quantities like velocity, force, or displacement.
Learn more about Vector with magnitude (absolute length) and phaseA **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves.
Learn more about Quantum Phase (direction). See ?.

By inversion, we mean that we add a minus sign to the realA real number is any number that can represent a distance along a continuous line, including all rational numbers (like fractions and integers) and irrational numbers (like or ). In other words, it’s any number that can appear on the number line. Real numbers exclude imaginary numbers, which involve .
Learn more about Real Number part of the complexA complex number is a number that has two parts: a real part and an imaginary part, written as , where . The real part behaves like ordinary numbers, while the imaginary part represents a direction perpendicular to the real axis on the complex plane. Complex numbers let us represent and calculate quantities involving square roots of negative numbers.
Learn more about Complex Number Amplitude.In quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
Learn more about Amplitude

Apparently, this flip of the sign has no effect on the magnitude. But it changes the direction of the Vector.A vector is a mathematical object that has both magnitude (size) and direction. It’s often represented as an arrow or as an ordered list of numbers (components) that describe its position in space, such as . Vectors are used to represent quantities like velocity, force, or displacement.
Learn more about Vector It changes the phaseA **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves.
Learn more about Quantum Phase. It marks the favorable Quantum State.A quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State

The heart of the OracleAn oracle is a black-box function that encodes information about a problem—typically deciding whether a given input satisfies some condition. It’s implemented as a quantum operation that can be queried in superposition, allowing a quantum algorithm to extract global properties of the function more efficiently than classical methods. Oracles are central to algorithms like Grover’s and Deutsch–Jozsa, where they guide the computation without revealing internal details.
Learn more about Oracle implementation is the Z-Gate.The Z-gate changes the phase of the state by (or radians) while leaving the state unchanged. This means the amplitude of gains a negative sign, effectively flipping its phase. It's a phase-flip operation that rotates the quantum state vector a half turn around the Z-axis of the Bloch sphere.
Learn more about Z-Gate The Z-GateThe Z-gate changes the phase of the state by (or radians) while leaving the state unchanged. This means the amplitude of gains a negative sign, effectively flipping its phase. It's a phase-flip operation that rotates the quantum state vector a half turn around the Z-axis of the Bloch sphere.
Learn more about Z-Gate rotates a the Quantum State VectorA quantum state vector is a mathematical object (usually denoted |ψ⟩) that fully describes the state of a quantum system. Its components give the probability amplitudes for finding the system in each possible basis state. The squared magnitude of each component gives the probability of measuring that corresponding outcome.
Learn more about Quantum State Vector of a QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit around the axis of the Bloch Sphere.The Bloch sphere is a geometric representation of a single qubit’s quantum state as a point on or inside a unit sphere. The north and south poles represent the classical states |0⟩ and |1⟩, while any other point corresponds to a superposition of them. Its position encodes the qubit’s relative phase and probability amplitudes, making it a visual tool for understanding quantum state evolution.
Learn more about Bloch Sphere

This rotation leaves the Quantum State Vector'sA quantum state vector is a mathematical object (usually denoted |ψ⟩) that fully describes the state of a quantum system. Its components give the probability amplitudes for finding the system in each possible basis state. The squared magnitude of each component gives the probability of measuring that corresponding outcome.
Learn more about Quantum State Vector distance from the poles (and thus its MeasurementIn quantum computing, measurement is the process of extracting classical information from a quantum state. It collapses a qubit’s superposition into one of its basis states (usually or ), with probabilities determined by the amplitudes of those states. After measurement, the qubit’s state becomes definite, destroying the original superposition.
Learn more about Measurement probabilities) unchanged, but reverses its Quantum Phase.A **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves.
Learn more about Quantum Phase See ?.

Marking the State

If the favorable state is is a basis state.
Learn more about , the Z-GateThe Z-gate changes the phase of the state by (or radians) while leaving the state unchanged. This means the amplitude of gains a negative sign, effectively flipping its phase. It's a phase-flip operation that rotates the quantum state vector a half turn around the Z-axis of the Bloch sphere.
Learn more about Z-Gate alone is sufficient. It multiplies the AmplitudeIn quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
Learn more about Amplitude of is a basis state.
Learn more about by and leaves is a basis state.
Learn more about unchanged.

? depicts the effect the Z-GateThe Z-gate changes the phase of the state by (or radians) while leaving the state unchanged. This means the amplitude of gains a negative sign, effectively flipping its phase. It's a phase-flip operation that rotates the quantum state vector a half turn around the Z-axis of the Bloch sphere.
Learn more about Z-Gate has on the AmplitudesIn quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
Learn more about Amplitude of a single Qubit.A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit

Marking the State

If we want to reverse the Quantum PhaseA **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves.
Learn more about Quantum Phase of is a basis state.
Learn more about instead, we must first select this state, then apply the Z-Gate,The Z-gate changes the phase of the state by (or radians) while leaving the state unchanged. This means the amplitude of gains a negative sign, effectively flipping its phase. It's a phase-flip operation that rotates the quantum state vector a half turn around the Z-axis of the Bloch sphere.
Learn more about Z-Gate and then unselect the state again.

For a single qubit, we can select the state is a basis state.
Learn more about with the X gateIn quantum computing, the NOT operation (also called the **X gate**) flips the state of a qubit: it turns (|0⟩) into (|1⟩) and (|1⟩) into (|0⟩). Mathematically, it’s represented by the Pauli-X matrix, which swaps the probability amplitudes of these basis states. On the Bloch sphere, it corresponds to a 180° rotation around the X-axis.
Learn more about Not Operation, since it swaps the AmplitudesIn quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
Learn more about Amplitude of is a basis state.
Learn more about and is a basis state.
Learn more about . The AmplitudeIn quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
Learn more about Amplitude of the Quantum StateA quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State is a basis state.
Learn more about thus becomes the AmplitudeIn quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
Learn more about Amplitude of is a basis state.
Learn more about when we apply the Z-GateThe Z-gate changes the phase of the state by (or radians) while leaving the state unchanged. This means the amplitude of gains a negative sign, effectively flipping its phase. It's a phase-flip operation that rotates the quantum state vector a half turn around the Z-axis of the Bloch sphere.
Learn more about Z-Gate immediately afterwards. And it becomes the AmplitudeIn quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
Learn more about Amplitude of is a basis state.
Learn more about again when we unselect the Quantum State.A quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State This means that when we apply the second X gateIn quantum computing, the NOT operation (also called the **X gate**) flips the state of a qubit: it turns (|0⟩) into (|1⟩) and (|1⟩) into (|0⟩). Mathematically, it’s represented by the Pauli-X matrix, which swaps the probability amplitudes of these basis states. On the Bloch sphere, it corresponds to a 180° rotation around the X-axis.
Learn more about Not Operation. Altogether, this effectively applies a Z-GateThe Z-gate changes the phase of the state by (or radians) while leaving the state unchanged. This means the amplitude of gains a negative sign, effectively flipping its phase. It's a phase-flip operation that rotates the quantum state vector a half turn around the Z-axis of the Bloch sphere.
Learn more about Z-Gate to is a basis state.
Learn more about .

How To Select A Computational Basis State

Don't do it manually

October 30, 2025•5 min

Advantage does not come from speed alone, but from clarity. And clarity begins with clarity with a small line of code.

? shows the code to mark the is a basis state.
Learn more about state.

amplification.py

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

from qiskit import QuantumCircuit

from basis_state_selector import SelectGate

 

qc = QuantumCircuit(1)

 

# Create |+> state

qc.h(0)

 

# select |0>, flip it to |1>

qc.append(SelectGate("0"), [0])

 

# flip phase of |1>

qc.z(0)

 

# unselect |0>, flip it back to |0>

qc.append(SelectGate("0"), [0])

Here, we
1. import the required functions from Qiskit again,
2. import the SelectGate function we developed in How To Select A Computational Basis State,
3. initialize a QuantumCircuit with a single Qubit,A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
   Learn more about Quantum Bit
4. apply a Hadamard OperatorThe Hadamard operator, often denoted H, is a single-qubit quantum gate that creates an equal superposition of and . In other words, It turns the states of the computational basis and into the states of the Fourier basis and .
   Learn more about Hadamard Operator on the qubit to put it into ,
5. select state is a basis state.
   Learn more about by flipping it to is a basis state.
   Learn more about
6. applies Z-GateThe Z-gate changes the phase of the state by (or radians) while leaving the state unchanged. This means the amplitude of gains a negative sign, effectively flipping its phase. It's a phase-flip operation that rotates the quantum state vector a half turn around the Z-axis of the Bloch sphere.
   Learn more about Z-Gate that flips the sign of the now is a basis state.
   Learn more about Amplitude.In quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
   Learn more about Amplitude
7. unselect is a basis state.
   Learn more about by flipping it back from is a basis state.
   Learn more about

? shows how the amplitudes change during this circuit.

Essentially, we create the state with a Global PhaseA global phase is a complex phase factor (like multiplying the whole quantum state by ) that changes how the state looks mathematically but not how it behaves physically. Since measurement probabilities depend only on relative phases between components, a global phase has no observable effect. So, the global phase doesn't change outcomes. It's physically meaningless and can be ignored.
Learn more about Global Phase

Marking a Specific State in Multi-Qubit Systems

When we have several qubits, follow the same pattern. First, we select the Quantum StateA quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State we want to mark. Therefore, we apply X gatesIn quantum computing, the NOT operation (also called the **X gate**) flips the state of a qubit: it turns (|0⟩) into (|1⟩) and (|1⟩) into (|0⟩). Mathematically, it’s represented by the Pauli-X matrix, which swaps the probability amplitudes of these basis states. On the Bloch sphere, it corresponds to a 180° rotation around the X-axis.
Learn more about Not Operation to all the QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit where the target Quantum StateA quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State is .

Next, we just want to reverse the Quantum PhaseA **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves.
Learn more about Quantum Phase of this one Quantum StateA quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State in which all QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit are is a basis state.
Learn more about . If we were to apply single-qubit Z-GatesThe Z-gate changes the phase of the state by (or radians) while leaving the state unchanged. This means the amplitude of gains a negative sign, effectively flipping its phase. It's a phase-flip operation that rotates the quantum state vector a half turn around the Z-axis of the Bloch sphere.
Learn more about Z-Gate to all Qubits,A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit we would select all states in which there is an odd number of is a basis state.
Learn more about s as depicted in ?.

To reverse only the AmplitudeIn quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
Learn more about Amplitude of this one state where all Qubits,A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit are is a basis state.
Learn more about , we use the Controlled Z Gate.A controlled-Z (CZ) gate is a two-qubit quantum gate that flips the phase of the second qubit (adds a minus sign) only if the first qubit is in the state . In matrix form, it leaves , , and unchanged, but turns into . It’s mainly used to create entanglement between qubits, just like the CNOT gate but with a phase flip instead of a bit flip.
Learn more about Controlled Z Gate In a two-qubit system, the Controlled Z GateA controlled-Z (CZ) gate is a two-qubit quantum gate that flips the phase of the second qubit (adds a minus sign) only if the first qubit is in the state . In matrix form, it leaves , , and unchanged, but turns into . It’s mainly used to create entanglement between qubits, just like the CNOT gate but with a phase flip instead of a bit flip.
Learn more about Controlled Z Gate applies a Quantum PhaseA **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves.
Learn more about Quantum Phase reversal only if both qubits are in the state is a basis state.
Learn more about as shown in ?. Since this operator is symmetric, each QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit can be considered as a control. The effect is the same. The AmplitudeIn quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
Learn more about Amplitude of the state is reversed.

When we have even more than two QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit , we use a multi-controlledA multi-controlled gate is a quantum logic gate that performs an operation on a target qubit only when several control qubits are all in a specific state (usually ). It generalizes the controlled-NOT (CNOT) gate, which has one control and one target. For example, a Toffoli gate is a 2-controlled NOT—it flips the target qubit only if both controls are 1.
Learn more about Multi-Controlled Gate Z-Gate.The Z-gate changes the phase of the state by (or radians) while leaving the state unchanged. This means the amplitude of gains a negative sign, effectively flipping its phase. It's a phase-flip operation that rotates the quantum state vector a half turn around the Z-axis of the Bloch sphere.
Learn more about Z-Gate This allows us to provide an arbitrary number of control QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit and only applies the phase flip on the Quantum StateA quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State where all the QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit are in is a basis state.
Learn more about .

At the end, we must not forget to unselect the basis state using a sequence of X gatesIn quantum computing, the NOT operation (also called the **X gate**) flips the state of a qubit: it turns (|0⟩) into (|1⟩) and (|1⟩) into (|0⟩). Mathematically, it’s represented by the Pauli-X matrix, which swaps the probability amplitudes of these basis states. On the Bloch sphere, it corresponds to a 180° rotation around the X-axis.
Learn more about Not Operation again.

Implementing the oracle

With these concepts at hand, implementing the OracleAn oracle is a black-box function that encodes information about a problem—typically deciding whether a given input satisfies some condition. It’s implemented as a quantum operation that can be queried in superposition, allowing a quantum algorithm to extract global properties of the function more efficiently than classical methods. Oracles are central to algorithms like Grover’s and Deutsch–Jozsa, where they guide the computation without revealing internal details.
Learn more about Oracle is straightforward. But since creating the OracleAn oracle is a black-box function that encodes information about a problem—typically deciding whether a given input satisfies some condition. It’s implemented as a quantum operation that can be queried in superposition, allowing a quantum algorithm to extract global properties of the function more efficiently than classical methods. Oracles are central to algorithms like Grover’s and Deutsch–Jozsa, where they guide the computation without revealing internal details.
Learn more about Oracle is only one step of Amplitude Amplification,Amplitude amplification is a quantum algorithmic technique that increases the probability of measuring desired outcomes by repeatedly applying a specific operator that amplifies their amplitude. It generalizes Grover’s search algorithm, working for any algorithm with a known success amplitude. After iterations, where is the original success probability, the target state's amplitude approaches , making it much more likely to be measured.
Learn more about Amplitude Amplification we want and need to work with our code later. Therefore, let's adhere to modularity and create a custom gate that creates an oracle for us.

How To Create Custom Operator With Qiskit

Stop Writing Quantum Spaghetti Code

October 29, 2025•5 min

Simply stacking quantum gates on one another is the fastest way to turn your code into a confusing mess

? provides the implementation of the OracleAn oracle is a black-box function that encodes information about a problem—typically deciding whether a given input satisfies some condition. It’s implemented as a quantum operation that can be queried in superposition, allowing a quantum algorithm to extract global properties of the function more efficiently than classical methods. Oracles are central to algorithms like Grover’s and Deutsch–Jozsa, where they guide the computation without revealing internal details.
Learn more about Oracle.

amplification.py

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

from qiskit import QuantumCircuit, QuantumRegister

from qiskit.circuit import Gate

from qiskit.circuit.library import ZGate

from basis_state_selector import SelectGate

 

 

def OracleGate(state_k: str) -> Gate:

"""

Create a Oracle that flips the phase of the basis state |state_k>.

 

Args:

state_k (str): MSB-encoded bitstring to tag. Example: '110' means |110> (MSB on the left).

 

Returns:

Gate: a quantum gate which tags |state_k>

 

Raises:

ValueError: for non-binary strings.

"""

if not isinstance(state_k, str):

raise ValueError("state_k must be a string, e.g. '0101'.")

if any(ch not in "01" for ch in state_k):

raise ValueError("state_k must contain only '0' and '1'.")

 

# A register to hold all our qubits

qr = QuantumRegister(len(state_k))

 

# A circuit to become our gate

qc = QuantumCircuit(qr, name=f"oracle({state_k})")

 

# select the given state

qc.append(SelectGate(state_k), qr)

 

# flip phase of |1..1>

qc.append(ZGate().control(num_ctrl_qubits=len(state_k) - 1), qr)

 

# unselect the given state

qc.append(SelectGate(state_k), qr)

 

return qc.to_gate(label=f"|{state_k}>")

In this code listing, we
1. import the required functions from Qiskit again,
2. import the SelectGate function we developed in How To Select A Computational Basis State,
3. verify the input. We throw a ValueError if the input is not a binary string,
4. create a QuantumRegister. This holds our QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
   Learn more about Quantum Bit and makes it easier to work with them,
5. create a QuantumCircuit that takes our QuantumRegister,
6. select the Basis StateA basis state in quantum computing is one of the fundamental states that form the building blocks of a quantum system’s state space. For a single qubit, the basis states are and ; any other qubit state is a superposition of these. In systems with multiple qubits, basis states are all possible combinations of s and s (e.g., , , , and ), forming an orthonormal basis for the system’s Hilbert space.
   Learn more about Basis State the OracleAn oracle is a black-box function that encodes information about a problem—typically deciding whether a given input satisfies some condition. It’s implemented as a quantum operation that can be queried in superposition, allowing a quantum algorithm to extract global properties of the function more efficiently than classical methods. Oracles are central to algorithms like Grover’s and Deutsch–Jozsa, where they guide the computation without revealing internal details.
   Learn more about Oracle is supposed to mark,
7. apply a multi-controlledA multi-controlled gate is a quantum logic gate that performs an operation on a target qubit only when several control qubits are all in a specific state (usually ). It generalizes the controlled-NOT (CNOT) gate, which has one control and one target. For example, a Toffoli gate is a 2-controlled NOT—it flips the target qubit only if both controls are 1.
   Learn more about Multi-Controlled Gate Z-GateThe Z-gate changes the phase of the state by (or radians) while leaving the state unchanged. This means the amplitude of gains a negative sign, effectively flipping its phase. It's a phase-flip operation that rotates the quantum state vector a half turn around the Z-axis of the Bloch sphere.
   Learn more about Z-Gate. Qiskit allows us to add controls to any of the standard gates. Here, we use the ZGate and append controls to it. We leave one QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
   Learn more about Quantum Bit as the target,
8. unselect the Basis StateA basis state in quantum computing is one of the fundamental states that form the building blocks of a quantum system’s state space. For a single qubit, the basis states are and ; any other qubit state is a superposition of these. In systems with multiple qubits, basis states are all possible combinations of s and s (e.g., , , , and ), forming an orthonormal basis for the system’s Hilbert space.
   Learn more about Basis State again,
9. turn the QuantumCircuit into a reusable Quantum GateA quantum gate is a basic operation that changes the state of one or more qubits, similar to how a logic gate operates on bits in classical computing. It uses unitary transformations, meaning it preserves the total probability (the state’s length in complex space). Quantum gates enable superposition and entanglement, allowing quantum computers to perform computations that classical ones cannot efficiently replicate.
   Learn more about Quantum Gate.

We can now use this gate in our overall Quantum CircuitA quantum circuit is a sequence of quantum gates applied to qubits, representing the operations in a quantum computation. Each gate changes the qubits’ state using quantum mechanics principles like superposition and entanglement. The final qubit states, when measured, yield the circuit’s computational result probabilistically.
Learn more about Quantum Circuit we started to implement in the previous post.

Amplitude Amplification

The Strange Art of Amplifying Success

October 22, 2025•5 min

Eight shells, one hidden gem, and a quantum trick that beats pure chance. Learn how quantum amplitude amplification uses simple geometry to bend probability.

? ...

amplification.py

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

from qiskit import QuantumCircuit

from qiskit.quantum_info import Statevector

 

# 3 qubits for 8 shells

n = 3

 

# initialize a quantum circuit

qc = QuantumCircuit(n)

 

# Put all 8 basis states into equal superposition

qc.h(range(n))

 

# append the oracle

qc.append(OracleGate("110"), range(n))

 

# Analyze the quantum state vector

psi = Statevector.from_instruction(qc)

 

# sample the counts

counts = psi.probabilities_dict()

The only change is the addition of the Oracle.

? depicts the overall circuit with the OracleAn oracle is a black-box function that encodes information about a problem—typically deciding whether a given input satisfies some condition. It’s implemented as a quantum operation that can be queried in superposition, allowing a quantum algorithm to extract global properties of the function more efficiently than classical methods. Oracles are central to algorithms like Grover’s and Deutsch–Jozsa, where they guide the computation without revealing internal details.
Learn more about Oracle for .

The OracleAn oracle is a black-box function that encodes information about a problem—typically deciding whether a given input satisfies some condition. It’s implemented as a quantum operation that can be queried in superposition, allowing a quantum algorithm to extract global properties of the function more efficiently than classical methods. Oracles are central to algorithms like Grover’s and Deutsch–Jozsa, where they guide the computation without revealing internal details.
Learn more about Oracle alone does not affect the MeasurementIn quantum computing, measurement is the process of extracting classical information from a quantum state. It collapses a qubit’s superposition into one of its basis states (usually or ), with probabilities determined by the amplitudes of those states. After measurement, the qubit’s state becomes definite, destroying the original superposition.
Learn more about Measurement outcomes. So, we still see that all states are equally likely in ?.

This is going to change in the next post, when we create the Diffusion OperatorThe diffusion operator (also called the Grover diffusion operator) is a unitary operation that inverts the amplitude of each state about the average amplitude. It’s used in Grover’s search algorithm to amplify the probability of the marked (target) state. Mathematically, it’s represented as , where is the uniform superposition state.
Learn more about Diffusion Operator.