Z-Gate

What You Get Is Not What You See

When you apply a 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 to a QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit, you expect to see a change when you measureIn 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 it. That is how classical logic works: flip a Bit,A bit (short for “binary digit”) is the smallest unit of data in computing, representing a value of either 0 or 1. It’s the fundamental building block of all digital information. Multiple bits combine to form larger units like bytes (8 bits) and encode more complex data such as numbers, text, or images.
Learn more about Binary Digit see the result.

This post is accompanied by a PDF file summarizing the key points.

But that's not how Quantum ComputingQuantum Computing is a different kind of computation that builds upon the phenomena of Quantum Mechanics.
Learn more about Quantum Computing works. When we look only at what we can measure immediately, we miss the operation that makes InterferenceInterference in quantum computing refers to the way probability amplitudes of quantum states combine—sometimes reinforcing each other (constructive interference) or canceling out (destructive interference). Quantum algorithms exploit this to amplify the probability of correct answers while suppressing incorrect ones. It’s a key mechanism that gives quantum computers their computational advantage.
Learn more about Interference possible.

There is one particular 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 that perfectly exposes this misconception. On paper, it seems pointless. When you measure after applying it, nothing looks different. Yet this single gate is one of the key tools that gives Quantum ComputingQuantum Computing is a different kind of computation that builds upon the phenomena of Quantum Mechanics.
Learn more about Quantum Computing its power. This is the Pauli -Gate.

The Pauli -Gate is a basic Quantum OperatorA quantum operator is a mathematical object that represents a physical action or measurement on a quantum state. It transforms one quantum state into another, often expressed as a matrix acting on a vector in Hilbert space. In quantum computing, operators correspond to quantum gates, which manipulate qubits according to the rules of linear algebra and quantum mechanics.
Learn more about Quantum Operator that flips 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 the Qubit'sA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit is a basis state.
Learn more about 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 while leaving the is a basis state.
Learn more about 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 unchanged. Mathematically, it multiplies is a basis state.
Learn more about by as depicted in ?.

So, apparently, the -Gate is not supposed to have an effect on a QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit in state is a basis state.
Learn more about . And it doesn't as depicted in ?. Of course, it doesn't. It only adds a minus sign to the is a basis state.
Learn more about component of the QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit. But that is . And .

Need some proof?

Anyone can just write that down.

OK. It doesn't change because 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 is and . So, to see what really happens, we need to test the -Gate on is a basis state.
Learn more about instead of is a basis state.
Learn more about .

But things don't even change when the QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit is in state is a basis state.
Learn more about .

Wait! That is different. Here, the -Gate turns is a basis state.
Learn more about into .

However, applying the -Gate on is a basis state.
Learn more about introduces what’s called 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 of . In Quantum Mechanics,Quantum mechanics is the branch of physics that describes the behavior of matter and energy at atomic and subatomic scales.
Learn more about Quantum Mechanics multiplying an entire 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 by a constant 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, such as , does not change anything you can measureIn 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.

And here's why. Probabilities in Quantum MechanicsQuantum mechanics is the branch of physics that describes the behavior of matter and energy at atomic and subatomic scales.
Learn more about Quantum Mechanics depend on the squared magnitude of the Quantum State'sA 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 Amplitudes.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 If every 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 multiplied by the same phase factor, the magnitudes remain the same.

That’s equivalent to rotating the Quantum Bit'sA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit overall phase by . But because this 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 applies equally to the whole Quantum State Vector,A 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 it has no observable effect. There is no change in 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 or InterferenceInterference in quantum computing refers to the way probability amplitudes of quantum states combine—sometimes reinforcing each other (constructive interference) or canceling out (destructive interference). Quantum algorithms exploit this to amplify the probability of correct answers while suppressing incorrect ones. It’s a key mechanism that gives quantum computers their computational advantage.
Learn more about Interference patterns if this QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit is isolated.

Geometrically, ? shows that the vector doesn't effectively change.

The -Gate turns 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 around the axis of the Bloch SphereThe 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. But since 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 is a basis state.
Learn more about lies on that axis, we can turn it as much as we want without actually changing it.

Finally, when "Nothing Happens"

What else can we try? How about a QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit in 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. That means, a QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit in a 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 of the Fourier BasisThe Fourier basis is a set of sine and cosine functions that can represent any periodic signal as a weighted sum of these functions. Each basis function corresponds to a specific frequency, capturing how much of that frequency is present in the signal. In essence, it’s the coordinate system for expressing signals in terms of their frequency components instead of time.
Learn more about Fourier Basis, such as .

Given a QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit in :

This time, the minus sign only applies to one of the two 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. We shifted 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 by radians. A half-turn on the Bloch SphereThe 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 as shown in ?.

It's my turn now to see it in action!

? prepares a QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit in , applies 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 on it, and compares both Quantum States.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

z_gate.py

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from numpy import allclose

from qiskit import QuantumCircuit

from qiskit.circuit.library import ZGate

from qiskit.quantum_info import Statevector

 

# start with one qubit in |0>

qc = QuantumCircuit(1)

 

# put it into |+>

qc.h(0)

 

# store the statevector of |+>

svplus = Statevector.from_instruction(qc)

 

# apply Z to |0>

qc.z(0)

 

# store the statevector of Z|+>

svZplus = Statevector.from_instruction(qc)

 

print(f"|+>: {svplus}")

print(f"Z|+>: {svZplus}")

print("|+> == Z|+> ?", allclose(svplus, svZplus))

 

counts_plus = svplus.probabilities_dict()

counts_Zplus = svZplus.probabilities_dict()

print(counts_plus)

print(counts_Zplus)

In this code listing, we
1. initialize a 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 with a single Qubit in state is a basis state.
   Learn more about .
2. apply the 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 to turn the qubit from is a basis state.
   Learn more about into
3. store the statevector of
4. 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
5. store the statevector of
6. compare the two Quantum State VectorsA 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
7. compare 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 results of these two Quantum StatesA 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

? shows the output of this compariosn.

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|+>: Statevector([0.70710678+0.j, 0.70710678+0.j], dims=(2,))

Z|+>: Statevector([ 0.70710678+0.j, -0.70710678+0.j], dims=(2,))

|+> == Z|+> ? False

{np.str_('0'): np.float64(0.4999999999999999), np.str_('1'): np.float64(0.4999999999999999)}

{np.str_('0'): np.float64(0.4999999999999999), np.str_('1'): np.float64(0.4999999999999999)}

As we see, the sign of the is a basis state.
Learn more about flipped. It changed 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 phase between is a basis state.
Learn more about and is a basis state.
Learn more about . But that's it. Applying 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 doesn't change 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.

That makes the gate look like it does nothing. It seems cosmetic, only.

---

The Power of a Sign

If you can’t see an effect, why bother?

Quantum MechanicsQuantum mechanics is the branch of physics that describes the behavior of matter and energy at atomic and subatomic scales.
Learn more about Quantum Mechanics is built on InterferenceInterference in quantum computing refers to the way probability amplitudes of quantum states combine—sometimes reinforcing each other (constructive interference) or canceling out (destructive interference). Quantum algorithms exploit this to amplify the probability of correct answers while suppressing incorrect ones. It’s a key mechanism that gives quantum computers their computational advantage.
Learn more about Interference. Every algorithm depends on it. InterferenceInterference in quantum computing refers to the way probability amplitudes of quantum states combine—sometimes reinforcing each other (constructive interference) or canceling out (destructive interference). Quantum algorithms exploit this to amplify the probability of correct answers while suppressing incorrect ones. It’s a key mechanism that gives quantum computers their computational advantage.
Learn more about Interference is the process where two 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 combine. If their signs match, they reinforce each other; but if one has a minus sign, they cancel out.

The magnitudes of the two components are unchanged. 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 probabilities are identical. But the relative sign between is a basis state.
Learn more about and is a basis state.
Learn more about has flipped. That is 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 change that determines whether paths add or cancel when the QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit interacts with others.

But once we apply another Hadamard, the difference becomes visible!

HZH-Routine

Taking Apart A Quantum Routine Doesn't Help To Understand It

November 12, 2025•5 min

The -Gate teaches one of the most counterintuitive lessons in Quantum Computing:Quantum Computing is a different kind of computation that builds upon the phenomena of Quantum Mechanics.
Learn more about Quantum Computing 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 is information. While 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 tells you how much of each 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 you have, 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 tells you how those 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 will interact when combined or entangled. Without control of 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 you cannot control Interference,Interference in quantum computing refers to the way probability amplitudes of quantum states combine—sometimes reinforcing each other (constructive interference) or canceling out (destructive interference). Quantum algorithms exploit this to amplify the probability of correct answers while suppressing incorrect ones. It’s a key mechanism that gives quantum computers their computational advantage.
Learn more about Interference and without Interference,Interference in quantum computing refers to the way probability amplitudes of quantum states combine—sometimes reinforcing each other (constructive interference) or canceling out (destructive interference). Quantum algorithms exploit this to amplify the probability of correct answers while suppressing incorrect ones. It’s a key mechanism that gives quantum computers their computational advantage.
Learn more about Interference Quantum AlgorithmsA quantum algorithm is a step-by-step computational procedure designed to run on a quantum computer, exploiting quantum phenomena such as superposition, entanglement, and interference to solve certain problems more efficiently than classical algorithms.
Learn more about Quantum Algorithm reduce to classical randomness.

So, the minus sign guides the Interference,Interference in quantum computing refers to the way probability amplitudes of quantum states combine—sometimes reinforcing each other (constructive interference) or canceling out (destructive interference). Quantum algorithms exploit this to amplify the probability of correct answers while suppressing incorrect ones. It’s a key mechanism that gives quantum computers their computational advantage.
Learn more about Interference redirecting it from constructive to destructive in the right place.